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8th Grade
Pythagorean Theorem, Distance & Midpoint
2016-01-15
www.njctl.org
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· Distance Formula
· Midpoints
Table of Contents
Click on a topic to go to that section
· Pythagorean Theorem
· Glossary & Standards
· Proofs
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· Distance Formula
· Midpoints
Table of Contents
Click on a topic to go to that section
· Pythagorean Theorem
· Glossary & Standards
· Proofs
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in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.
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Proofs
Click to return to the table of contents
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Introduction to Proofs
Lab: Introduction to Proofs
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Introduction to Proofs
Lab: Introduction to Proofs
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Have the students brainstorm the steps required to make a sandwich in small
groups (2-4 students per group).The questions can be completed either in
their groups or as a class discussion (located on the next 6 slides).
The questions in this lab address MP.3:Ask: What are some possibilities here?How can you prove that your answer is
correct?
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1 Determine which type of sandwich you would like to start. Choose only one.
A Peanut Butter & JellyB Ham and CheeseC FluffernutterD Tuna melt BLTE Chicken Salad BLT
F Peanut Butter & Banana
G Bologna and CheeseH Grilled Cheese
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1 Determine which type of sandwich you would like to start. Choose only one.
A Peanut Butter & JellyB Ham and CheeseC FluffernutterD Tuna melt BLTE Chicken Salad BLT
F Peanut Butter & Banana
G Bologna and CheeseH Grilled Cheese
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Fill in the blank on the next page with the winner of this
poll.
If you wanted to, you could make the class prepare
your lunch sandwich that day and adjust the wording
on the next few slides.
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2) Fill in the blank with the winner of the class poll.Everyday for lunch, you make yourself a sandwich. The sandwich that you are making today is
______________________________________.
3) What steps do you need to take in order to make your sandwich? List all of the thoughts of your group in the space provided on the Lab WS.
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2) Fill in the blank with the winner of the class poll.Everyday for lunch, you make yourself a sandwich. The sandwich that you are making today is
______________________________________.
3) What steps do you need to take in order to make your sandwich? List all of the thoughts of your group in the space provided on the Lab WS.
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Split the students up into groups of 2-4.
Give them about 5-10 minutes to brainstorm their ideas to make the sandwich.
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4) List any additional thoughts provided by the rest of the groups in the class.
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4) List any additional thoughts provided by the rest of the groups in the class.
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This is where you list all of the ideas that your students come up with. The wording of the question corresponds to the Lab WS for the students.
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5) List all of the thoughts contributed by your group and the rest of the class in the chronological order decided upon by the class.
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5) List all of the thoughts contributed by your group and the rest of the class in the chronological order decided upon by the class.
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chronologically list all of the student thoughts.
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6) Was the sandwich made correctly? Why or why not? If not, what steps do you need to add?
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6) Was the sandwich made correctly? Why or why not? If not, what steps do you need to add?
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any additional steps required.
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7) Write down the revised list all of the thoughts contributed by your group and the rest of the class in the chronological order.
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7) Write down the revised list all of the thoughts contributed by your group and the rest of the class in the chronological order.
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chronologically list the revised list of the student thoughts.
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Proofs
A proof involves writing reasoned, logical explanations that use definitions, algebraic properties, postulates, and previously proven theorems to arrive at a conclusion.
A postulate is a property that is accepted without proof, usually because one can easily understand why it is true. One of the classic postulates of Geometry states that "Through any two points, there is exactly one line".
AB
AB
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ProofsThe lab that you completed about making a sandwich has the same general steps as a proof. You brainstorm and write down various ideas based on what you know, and organize them into the proper order that will lead you to the final conclusion.
You can also use proofs when solving algebraic equations. As you write each step/idea, you also write down the definition, property, postulate, etc. that justifies your step/idea. The example shown below is written as a two-column proof.
Given: 3x + 2 = 14 Prove: x = 4
Note: The first step to solving a proof is always "Given" to you.
Statements Reasons1. 3x + 2 = 14 1. Given
2. 3x = 12 2. Subtraction
3. x = 4 3. Division
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ProofsWhen writing two-column proofs, you can also combine the rules of algebra and properties of geometry.
Example:Given: m∠A = (4x + 3)°, m∠B = (5x + 4)°, m∠C = (x + 23)°Prove: ΔABC is an acute triangle
(5x + 4)°
(4x + 3)° (x + 23)°A
B
C
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ProofsGiven: m∠A = (4x + 3)°, m∠B = (5x + 4)°, m∠C = (x + 23)°Prove: ΔABC is an acute triangle
(5x + 4)°
(4x + 3)° (x + 23)°A
B
CStatements Reasons
1. m∠A = (4x + 3)°, m∠B = (5x + 4)°, m∠C = (x + 23)° 1. Given
2. m∠A + m∠B + m∠C = 180° 2. All of the interior angles of a triangle sum to 180°
3. 4x + 3 + 5x + 4 + x + 23 = 180 3. Substitution 4. 10x + 30 = 180 4. Combine Like Terms5. 10x = 150 5. Subtraction6. x = 15 6. Division7. m∠A = 4(15) + 3 = 63°, m∠B = 5(15) + 4 = 79° m∠C = (15) + 23 = 38° 7. Substitution
8. ΔABC is an acute triangle 8. Definition of an acute Δ
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Proofs
Example: Write a two-column proof.
Given: 2(x + 4) + 4(x - 5) = -42Prove: x = -5
Statements Reasons
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Proofs
Example: Write a two-column proof.
Given: 2(x + 4) + 4(x - 5) = -42Prove: x = -5
Statements Reasons
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Statements1. 2(x + 4) + 4(x - 5) = -422. 2x + 8 + 4x - 20 = -423. 6x - 12 = -42
4. 6x = -305. x = -5
Reasons1. Given2. Distributive Property3. Combine Like Terms Simplify4. Addition5. Division
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ProofsExample:Given: m∠D = (4x + 7)°, m∠E = (9x + 5)°, m∠F = (2x + 3)°Prove: ΔDEF is an obtuse Δ
(9x + 5)°
(4x + 7)° (2x + 3)°D
E
F
Statements Reasons
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ProofsExample:Given: m∠D = (4x + 7)°, m∠E = (9x + 5)°, m∠F = (2x + 3)°Prove: ΔDEF is an obtuse Δ
(9x + 5)°
(4x + 7)° (2x + 3)°D
E
F
Statements Reasons
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Statements1. m∠D = (4x + 7)°, m∠E = (9x + 5)°, m∠F = (2x + 3)°2. m∠D + m∠E + m∠F = 180°
3. 4x + 7 + 9x + 5 + 2x + 3 = 180
4. 15x + 15 = 1805. 15x = 1656. x = 117. m∠D = 4(11) + 7 = 51°, m∠E = 9(11) + 5 = 104°, m∠F = 2(11) + 3 = 25°8. ΔDEF is an obtuse Δ
Reasons1. Given2. The interior angles of a triangle sum to 1803. Substitution4. Combine Like Terms5. Subtraction6. Division
7. Substitution
8. Definition of an obtuse Δ
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Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
ProofsThe next 10 Response Questions are all based on the problem below. If needed, fill in the two-column proof as you answer each question.
Statements Reasons
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2 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the first statement?
A -6x - 18 + 3x + 15 = -18
B -6x + 18 + 3x - 15 = -18
C -6(x - 3) + 3(x - 5) = -18
D -6x - 18 + 3x - 15 = -18
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2 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the first statement?
A -6x - 18 + 3x + 15 = -18
B -6x + 18 + 3x - 15 = -18
C -6(x - 3) + 3(x - 5) = -18
D -6x - 18 + 3x - 15 = -18
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C
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3 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the first reason?
A Distributive Property
B Addition
C Multiplication
D Given
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3 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the first reason?
A Distributive Property
B Addition
C Multiplication
D Given
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D
Statements
1. -6(x - 3) + 3(x - 5) = -18
Reasons
1. Given
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4 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the second statement?
A -6x - 18 + 3x + 15 = -18
B -6x + 18 + 3x - 15 = -18
C -6(x - 3) + 3(x - 5) = -18
D -6x - 18 + 3x - 15 = -18
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4 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the second statement?
A -6x - 18 + 3x + 15 = -18
B -6x + 18 + 3x - 15 = -18
C -6(x - 3) + 3(x - 5) = -18
D -6x - 18 + 3x - 15 = -18
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B
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5 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the second reason?
A Distributive Property
B Addition
C Multiplication
D Given
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5 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the second reason?
A Distributive Property
B Addition
C Multiplication
D Given
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A
Statements
1. -6(x - 3) + 3(x - 5) = -182. -6x + 18 + 3x - 15 = -18
Reasons
1. Given2. Distributive Property
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6 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the third statement?
A -3x - 3 = -18
B 9x - 3 = -18
C 9x + 3 = -18
D -3x + 3 = -18
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6 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the third statement?
A -3x - 3 = -18
B 9x - 3 = -18
C 9x + 3 = -18
D -3x + 3 = -18
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D
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7 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the third reason?
A Simplify or Combine Like Terms
B Addition
C Division
D Subtraction
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7 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the third reason?
A Simplify or Combine Like Terms
B Addition
C Division
D Subtraction
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A
Statements
1. -6(x - 3) + 3(x - 5) = -182. -6x + 18 + 3x - 15 = -18
3. -3x + 3 = -18
Reasons
1. Given2. Distributive Property3. Simplify orCombine Like Terms
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8 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the fourth statement?
A -3x = -15
B -3x = -21
C 9x = -15
D 9x = -21
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8 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the fourth statement?
A -3x = -15
B -3x = -21
C 9x = -15
D 9x = -21
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B
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9 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the fourth reason?
A Simplify or Combine Like Terms
B Addition
C Division
D Subtraction
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9 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the fourth reason?
A Simplify or Combine Like Terms
B Addition
C Division
D Subtraction
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D
Statements
1. -6(x - 3) + 3(x - 5) = -182. -6x + 18 + 3x - 15 = -18
3. -3x + 3 = -18
4. -3x = -21
Reasons
1. Given2. Distributive Property3. Simplify orCombine Like Terms4. Subtraction
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10 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the fifth statement?
A x = 7
B x = 5
C x = -
D x = -
7 3 5 3
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10 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the fifth statement?
A x = 7
B x = 5
C x = -
D x = -
7 3 5 3
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A
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11 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the fifth reason?
A Simplify or Combine Like Terms
B Addition
C Division
D Subtraction
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11 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7
What is the fifth reason?
A Simplify or Combine Like Terms
B Addition
C Division
D Subtraction
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Ans
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C
Statements
1. -6(x - 3) + 3(x - 5) = -182. -6x + 18 + 3x - 15 = -18
3. -3x + 3 = -18
4. -3x = -215. x = 7
Reasons
1. Given2. Distributive Property3. Simplify orCombine Like Terms4. Subtraction5. Division
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Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFG.
ProofsThe next 8 Response Questions are all based on the problem below. If needed, fill in the two-column proof as you answer each question.
(6x + 7)° (11x + 3)°
A B
C
D
(7x - 3)°
E
FG
Statements1. ∠ABC and ∠CBD are supplementary2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. _____________________2. _____________________3. _____________________4. _____________________5. _____________________6. _____________________7. _____________________
8. _____________________
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12 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the first reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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12 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the first reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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F
Statements1. ∠ABC and ∠CBD are supp.2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. Given2. ______________3. ______________4. ______________5. ______________6. ______________7. ______________8. ______________
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13 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the second reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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13 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the second reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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A
Statements1. ∠ABC and ∠CBD are supp.2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. Given2. Def. of supplementary ∠s
3. ______________4. ______________5. ______________6. ______________7. ______________8. ______________
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14 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the third reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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14 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the third reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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Ans
wer
C
Statements1. ∠ABC and ∠CBD are supp.2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. Given2. Def. of supplementary ∠s
3. Substitution4. ______________5. ______________6. ______________7. ______________8. ______________
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15 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the fourth reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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15 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the fourth reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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H
Statements1. ∠ABC and ∠CBD are supp.2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. Given2. Def. of supplementary ∠s
3. Substitution4. Combine Like Terms5. ______________6. ______________7. ______________8. ______________
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16 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the fifth reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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16 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the fifth reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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Ans
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G
Statements1. ∠ABC and ∠CBD are supp.2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. Given2. Def. of supplementary ∠s
3. Substitution4. Combine Like Terms5. Subtraction6. ______________7. ______________8. ______________
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17 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the sixth reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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17 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the sixth reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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Ans
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D
Statements1. ∠ABC and ∠CBD are supp.2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. Given2. Def. of supplementary ∠s
3. Substitution4. Combine Like Terms5. Subtraction6. Division7. ______________8. ______________
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18 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the seventh reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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18 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the seventh reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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C
Statements1. ∠ABC and ∠CBD are supp.2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. Given2. Def. of supplementary ∠s
3. Substitution4. Combine Like Terms5. Subtraction6. Division7. Substitution8. ______________
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19 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the seventh reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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19 Given: ∠ABC and ∠CBD are supplementaryProve: ∠ABC ≅ ∠EFGWhat is the seventh reason?A Definition of supplementary angles
B Addition
C Substitution
D Division
E Definition of congruent angles
F Given
G Subtraction
H Combine Like Terms
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Ans
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E
Statements1. ∠ABC and ∠CBD are supp.2. m∠ABC + m∠CBD = 180°3. 6x + 7 + 11x + 3 = 1804. 17x + 10 = 1805. 17x = 1706. x = 107. m∠ABC = 6(10) + 7 = 67°, m∠EFG = 7(10) - 3 = 67° 8. ∠ABC ≅ m∠EFG
Reasons1. Given2. Def. of supplementary ∠s
3. Substitution4. Combine Like Terms5. Subtraction6. Division7. Substitution8. Def. of ≅ angles
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Pythagorean Theorem
Click to return to the table of contents
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Pythagorean Theorem
Pythagorean theorem is used for right triangles. It was first known in ancient Babylon and Egypt beginning about 1900 B.C. However, it was not widely known until Pythagoras stated it.
Pythagoras lived during the 6th century B.C. on the island of Samos in the Aegean Sea. He also lived in Egypt, Babylon, and southern Italy. He was a philosopher and a teacher.
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Legs
- Opposite the right angle- Longest of the 3 sides
- 2 sides that form the right angle
click to reveal
click to reveal
Labels
ca
b
Hypotenuseclick to reveal
click to reveal
for a right triangle
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In a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of the hypotenuse (c).
a2 + b2 = c2
· Moving of squares
· Water demo
Click on the links below to see several
animations of the proof
· Move slider to show c2
Pythagorean Theorem Proofs
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Lab: Proof of the Pythagorean Theorem
Proof of the Pythagorean Theorem
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Lab: Proof of the Pythagorean Theorem
Proof of the Pythagorean Theorem
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her N
otes
Have the students complete parts #1-3 in their small groups (2-4 students per group).
The "Follow-up questions" can be completed either in their groups or as a class discussion (located on the next 4 slides).
Slide 43 / 145
Follow-up Questions:
11. How is the area of the printed square on page 1 related to the area of the printed square on page 2? Explain how you reached that conclusion.
12. How are the 4 right triangles that you cut out at the beginning of this lab related to each other? Explain how you reached that conclusion.
Slide 43 (Answer) / 145
Follow-up Questions:
11. How is the area of the printed square on page 1 related to the area of the printed square on page 2? Explain how you reached that conclusion.
12. How are the 4 right triangles that you cut out at the beginning of this lab related to each other? Explain how you reached that conclusion.
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Mat
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Question #12: MP 6 & MP7
Slide 44 / 145
Follow-up Questions (cont'd):
13. How are the areas that you found in question #6 & question #10 related to each other? Explain how you reached that conclusion.
14. What algebraic expression represents the side length of the printed squares. Explain how you reached that conclusion.
Slide 44 (Answer) / 145
Follow-up Questions (cont'd):
13. How are the areas that you found in question #6 & question #10 related to each other? Explain how you reached that conclusion.
14. What algebraic expression represents the side length of the printed squares. Explain how you reached that conclusion.
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Question #13: MP6 & MP7
Question #14: MP2 & MP3
Slide 45 / 145
Follow-up Questions (cont'd):
16. Using the arrangement of the shapes from question #5, write an algebraic expression to represent the area of the entire figure.
15. Multiply these side lengths together and simplify the expression.
Slide 45 (Answer) / 145
Follow-up Questions (cont'd):
16. Using the arrangement of the shapes from question #5, write an algebraic expression to represent the area of the entire figure.
15. Multiply these side lengths together and simplify the expression.
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Question #15: MP2
Question #16: MP2
Slide 46 / 145
Follow-up Questions (cont'd):
17. Set the expressions from question #15 & question #16 equal to one another and simplify the equation.
Slide 46 (Answer) / 145
Follow-up Questions (cont'd):
17. Set the expressions from question #15 & question #16 equal to one another and simplify the equation.
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Question #17: MP2
Slide 47 / 145
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer
Missing Hypotenuse
Write Equation
Substitute in numbers
Square numbers
Add
Find the Square Root
Label Answer
How to use the formula to find missing sides.
Pythagorean Theorem
Slide 48 / 145
a2 + b2 = c2
52 + b2 = 152
25 + b2 = 225
-25 -25
b2 = 200
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer5 ft
15 ft
Pythagorean Theorem
Slide 48 (Answer) / 145
a2 + b2 = c2
52 + b2 = 152
25 + b2 = 225
-25 -25
b2 = 200
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer5 ft
15 ft
Pythagorean Theorem
[This object is a pull tab]
Mat
h Pr
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eMP.6: Attend to precision
Ask: How do you know that your answer is accurate?
To get the answer: Talk about finding the perfect squares before and after the
radicand in the problem.
What labels could you use?
Slide 49 / 145
9 in18 in
a2 + b2 = c2
92 + b2 = 182
81 + b2 = 324
-81 -81
b2 = 243
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer
Pythagorean Theorem
Slide 50 / 145
4 in
7 in
a2 + b 2 = c 2
42 + 7 2 = c 2
16 + 49 = c 2
65 = c 2
Missing Hypotenuse
Write Equation
Substitute in numbers
Square numbers
Add
Find the Square Root &Label Answer
Pythagorean Theorem
Slide 51 / 145
20 What is the length of the third side?
4
7x
Slide 51 (Answer) / 145
20 What is the length of the third side?
4
7x
[This object is a pull tab]
Ans
wer 42 + 72 = x2
16 + 49 = x 2
65 = x2
Slide 52 / 145
21 What is the length of the third side?
41x
15
Slide 52 (Answer) / 145
21 What is the length of the third side?
41x
15
[This object is a pull tab]
Ans
wer 152 + 412 = x2
225 + 1681 = x 2
1906 = x 2
Slide 53 / 145
22 What is the length of the third side?
7
x4
Slide 53 (Answer) / 145
22 What is the length of the third side?
7
x4
[This object is a pull tab]
Ans
wer x2 + 42 = 72
x2 + 16 = 49 x2 = 33
Slide 54 / 145
23 What is the length of the third side?
3
4
x
Slide 54 (Answer) / 145
23 What is the length of the third side?
3
4
x
[This object is a pull tab]
Ans
wer 32 + 42 = x2
9 + 16 = x2
25 = x2
5 = x
Slide 55 / 145
3
4
5
There are combinations of whole numbers that work in the Pythagorean Theorem. These sets of numbers are known as Pythagorean Triples.
3-4-5 is the most famous of the triples. If you recognize the sides of the triangle as being a triple (or multiple of one), you won't need a calculator!
Pythagorean Triples
Slide 56 / 145
Trip
les
12 = 1 112 = 121 212 = 44122 = 4 122 = 144 222 = 48432 = 9 132 = 169 232 = 52942 = 16 142 = 196 242 = 57652 = 25 152 = 225 252 = 62562 = 36 162 = 256 262 = 67672 = 49 172 = 289 272 = 72982 = 64 182 = 324 282 = 78492 = 81 192 = 361 292 = 841102 = 100 202 = 400 302 = 900
Can you find any other Pythagorean Triples?
Use the list of squares to see if any other triples work.
Pythagorean Triples
Slide 56 (Answer) / 145
Trip
les
12 = 1 112 = 121 212 = 44122 = 4 122 = 144 222 = 48432 = 9 132 = 169 232 = 52942 = 16 142 = 196 242 = 57652 = 25 152 = 225 252 = 62562 = 36 162 = 256 262 = 67672 = 49 172 = 289 272 = 72982 = 64 182 = 324 282 = 78492 = 81 192 = 361 292 = 841102 = 100 202 = 400 302 = 900
Can you find any other Pythagorean Triples?
Use the list of squares to see if any other triples work.
Pythagorean Triples
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Ans
wer
&
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Pythagorean Triples3 - 4 - 5 5 - 12 - 137 - 24 - 25 8 - 15 - 17
Multiples of these combinations work too!
MP.8: Look for and express regularity in repeated reasoning.
Ask: Is it true every time?What concepts that have we learned
before were useful in solving this problem?
Could this problem help you solve another problem?
Slide 57 / 145
24 What is the length of the third side?
6
8
Slide 57 (Answer) / 145
24 What is the length of the third side?
6
8 [This object is a pull tab]
Ans
wer 62 + 82 = x2 OR
36+ 64 = x2
100 = x2
10 = x
Slide 58 / 145
25 What is the length of the third side?
513
Slide 58 (Answer) / 145
25 What is the length of the third side?
513
[This object is a pull tab]
Ans
wer
52 + x2 = 132
25+ x2 = 169 x2 = 144 x = 12
OR
5-12-13
Slide 59 / 145
26 What is the length of the third side?
48
50
Slide 59 (Answer) / 145
26 What is the length of the third side?
48
50
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Ans
wer x2 + 482 = 502 OR
x2 + 2304 = 2500 x2 = 196 x = 14
Slide 60 / 145
27 The legs of a right triangle are 7.0 and 3.0, what is the length of the hypotenuse?
Slide 60 (Answer) / 145
27 The legs of a right triangle are 7.0 and 3.0, what is the length of the hypotenuse?
[This object is a pull tab]
Ans
wer 32 + 72 = x2
9 + 49 = x2
58 = x2
Slide 61 / 145
28 The legs of a right triangle are 2.0 and 12, what is the length of the hypotenuse?
Slide 61 (Answer) / 145
28 The legs of a right triangle are 2.0 and 12, what is the length of the hypotenuse?
[This object is a pull tab]
Ans
wer
22 + 122 = x2 4 + 144 = x2
148 = x2
Slide 62 / 145
29 The hypotenuse of a right triangle has a length of 4.0 and one of its legs has a length of 2.5. What is the length of the other leg?
Slide 62 (Answer) / 145
29 The hypotenuse of a right triangle has a length of 4.0 and one of its legs has a length of 2.5. What is the length of the other leg?
[This object is a pull tab]
Ans
wer x2 + 2.52 = 42
x2 + 6.25 = 16 x2 = 9.75
Slide 63 / 145
30 The hypotenuse of a right triangle has a length of 9.0 and one of its legs has a length of 4.5. What is the length of the other leg?
Slide 63 (Answer) / 145
30 The hypotenuse of a right triangle has a length of 9.0 and one of its legs has a length of 4.5. What is the length of the other leg?
[This object is a pull tab]
Ans
wer x2 + 4.52 = 92
x2 + 20.25 = 81 x2 = 60.75
Slide 64 / 145
This is a great problem and draws on a lot of what we've learned.
Try it in your groups. Then we'll work on it step by step together by asking questions that break the problem into pieces.
From PARCC EOY sample test calculator #1
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
A
B
CD
810 10
Slide 65 / 145
31 What have we learned that will help solve this problem?A Pythagorean Theorem
B Pythagorean Triples
C Distance Formula
D A and B only
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
A
B
CD
810 10
Slide 65 (Answer) / 145
31 What have we learned that will help solve this problem?A Pythagorean Theorem
B Pythagorean Triples
C Distance Formula
D A and B only
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
A
B
CD
810 10
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Ans
wer
D A and B only
Slide 66 / 145
A
B
CD
810 10
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
First, notice that we have two right triangles (perpendicular lines make right angles). The triangles are outlined red & blue in the diagram below.
Slide 67 / 145
32 What is the length of the 3rd side in the red triangle?A 3 cmB 6 cmC 9 cmD 13.45 cm
A
B
CD
810 10
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
Slide 67 (Answer) / 145
32 What is the length of the 3rd side in the red triangle?A 3 cmB 6 cmC 9 cmD 13.45 cm
A
B
CD
810 10
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
[This object is a pull tab]
Ans
wer
B
a2 + 82 = 102
a2 + 64 = 100a2 = 36a = 6
or 2(3-4-5) = 6-8-10, so a = 6
Slide 68 / 14533 How is AD related to CD?
A AD > CD
B AD < CD
C AD = CDD not enough information to relate these segments
6A
B
CD
810 10
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
Slide 68 (Answer) / 14533 How is AD related to CD?
A AD > CD
B AD < CD
C AD = CDD not enough information to relate these segments
6A
B
CD
810 10
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
[This object is a pull tab]
Ans
wer
C AD = CD
Two right triangles are equal, so their corresponding sides are equal. Also, if you use Pythagorean Theorem again to find CD, it will also equal 6.
Slide 69 / 145
34 What is the length of AC?
A
B
CD
810 10
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
Slide 69 (Answer) / 145
34 What is the length of AC?
A
B
CD
810 10
In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters.
What is the length of AC?
[This object is a pull tab]
Ans
wer 6 + 6
12
Slide 70 / 145
Converse of the Pythagorean TheoremIf a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c2 = a2 + b2, then the triangle is a right triangle.
If c2 ≠ a2 + b2, then the triangle is not a right triangle. This is the Converse of the Pythagorean Theorem.
b = 4 ft
c = 5 fta = 3 ft
Slide 71 / 145
Converse of the Pythagorean Theorem
In other words, you can check to see if a triangle is a right triangle by seeing if the Pythagorean Theorem is true.
Test the Pythagorean Theorem. If the final equation is true, then the triangle is right. If the final equation is false, then the triangle is not right.
Slide 72 / 145
Is it a Right Triangle?
Write Equation
Plug in numbers
Square numbers
Simplify both sides
Are they equal?
8 in, 17 in, 15 in
a2 + b2 = c2
82 + 152 = 172
64 + 225 = 289
289 = 289
Yes!
Converse of the Pythagorean Theorem
Slide 72 (Answer) / 145
Is it a Right Triangle?
Write Equation
Plug in numbers
Square numbers
Simplify both sides
Are they equal?
8 in, 17 in, 15 in
a2 + b2 = c2
82 + 152 = 172
64 + 225 = 289
289 = 289
Yes!
Converse of the Pythagorean Theorem
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MP.1: Make sense of problems and persevere in solving them
MP.8: Look for and express regularity in repeated reasoning.
Ask: What is this problem asking? (MP.1)How could you start this problem? (MP.1)
What concepts that we have learned before were useful in solving this
problem? (MP.8) What generalizations can you make?
(MP.8)Answer: If the numbers are a
Pythagorean Triple, (or multiple of one) then it's a right triangle.
Slide 73 / 145
35 Is the triangle a right triangle?
YesNo
8 ft
10 ft6 ft
Slide 73 (Answer) / 145
35 Is the triangle a right triangle?
YesNo
8 ft
10 ft6 ft
[This object is a pull tab]
Ans
wer
62 + 82 = 102 36 + 64 = 100 100 = 100
YES
OR
Pythagorean Triple3-4-5
Slide 74 / 145
36 Is the triangle a right triangle?
YesNo
30 ft
24 ft36 ft
Slide 74 (Answer) / 145
36 Is the triangle a right triangle?
YesNo
30 ft
24 ft36 ft
[This object is a pull tab]
Ans
wer 242 + 302 = 362
576 + 900 = 1296 1476 = 1296
NO
Slide 75 / 145
37 Is the triangle a right triangle?
YesNo
10 in.
8 in. 12 in.
Slide 75 (Answer) / 145
37 Is the triangle a right triangle?
YesNo
10 in.
8 in. 12 in.
[This object is a pull tab]
Ans
wer 82 + 102 = 122
64 + 100 = 144 164 = 196
NO
Slide 76 / 145
38 Is the triangle a right triangle?
YesNo
5 ft13 ft
12 ft
Slide 76 (Answer) / 145
38 Is the triangle a right triangle?
YesNo
5 ft13 ft
12 ft
[This object is a pull tab]
Ans
wer Yes - Pythagorean Triple!
5-12-13
Slide 77 / 145
39 Can you construct a right triangle with three lengths of wood that measure 7.5 in, 18 in and 19.5 in?
YesNo
Slide 77 (Answer) / 145
39 Can you construct a right triangle with three lengths of wood that measure 7.5 in, 18 in and 19.5 in?
YesNo
[This object is a pull tab]
Ans
wer 7.52 + 182 = 19.52
56.25 + 324 = 380.25 380.25 = 380.25
YES
Slide 78 / 145
Steps to Pythagorean Theorem Application Problems.
1. Draw a right triangle to represent the situation.2. Solve for unknown side length.3. Round to the nearest tenth.
Applications of Pythagorean Theorem
Slide 78 (Answer) / 145
Steps to Pythagorean Theorem Application Problems.
1. Draw a right triangle to represent the situation.2. Solve for unknown side length.3. Round to the nearest tenth.
Applications of Pythagorean Theorem
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The examples in this lesson (next 6 slides) address
MP.4: Model with mathematicsMP.5: Use appropriate tools strategically.
Ask: What do you already know about solving this problem? (MP.4)
What connections do you see between this problem and Pythagorean Theorem?
(MP.4)How could you use manipulatives or a drawing to show your thinking? (MP.5)
Slide 79 / 145
5
10
5 10
-5
-5x
-10
0-10
yWork with your partners to complete:To get from his high school to his home, Jamal travels 5.0 miles east and then 4.0 miles north. When Sheila goes to her home from the same high school, she travels 8.0 miles east and 2.0 miles south. What is the measure of the shortest distance, to the nearest tenth of a mile, between Jamal's home and Sheila's home?
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Applications of Pythagorean Theorem
Slide 79 (Answer) / 145
5
10
5 10
-5
-5x
-10
0-10
yWork with your partners to complete:To get from his high school to his home, Jamal travels 5.0 miles east and then 4.0 miles north. When Sheila goes to her home from the same high school, she travels 8.0 miles east and 2.0 miles south. What is the measure of the shortest distance, to the nearest tenth of a mile, between Jamal's home and Sheila's home?
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Applications of Pythagorean Theorem
[This object is a pull tab]
Ans
wer 62 + 32 = x2
36 + 9 = x2
45 = x2
6.7 = x
Slide 80 / 145
Work with your partners to complete:
A straw is placed into a rectangular box that is 3 inches by 4 inches by 8 inches, as shown in the accompanying diagram. If the straw fits exactly into the box diagonally from the bottom left front corner to the top right back corner, how long is the straw, to the nearest tenth of an inch?
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Applications of Pythagorean Theorem
Slide 80 (Answer) / 145
Work with your partners to complete:
A straw is placed into a rectangular box that is 3 inches by 4 inches by 8 inches, as shown in the accompanying diagram. If the straw fits exactly into the box diagonally from the bottom left front corner to the top right back corner, how long is the straw, to the nearest tenth of an inch?
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Applications of Pythagorean Theorem
[This object is a pull tab]
Ans
wer
32 + 42 = 52
Pythagorean Triple c = 5
c2 + d2 = e2 52 + 82 = e2
89 = e2
9.4 = e
a bc
de
Slide 81 / 145
The Pythagorean Theorem can be applied to 3 Dimensional Figures
In this figure:
a = slant height (height of triangular face)
b = 1/2 base length (from midpoint of side of base to center of the base of the pyramid)
h = height of pyramid
Applications of Pythagorean Theorem
Slide 82 / 145
A right triangle is formed between the three lengths.
If you know two of the measurements, you can calculate the third.
EXAMPLE:Find the slant height of a pyramidwhose height is 5 cm and whosebase has a length of 8cm.
Applications of Pythagorean Theorem
Slide 82 (Answer) / 145
A right triangle is formed between the three lengths.
If you know two of the measurements, you can calculate the third.
EXAMPLE:Find the slant height of a pyramidwhose height is 5 cm and whosebase has a length of 8cm.
Applications of Pythagorean Theorem
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Ans
wer
Slide 83 / 145
Find the slant height of the pyramid whose base length is 10 cm and height is 12 cm. Label the diagram with the measurements.
Applications of Pythagorean Theorem
Slide 83 (Answer) / 145
Find the slant height of the pyramid whose base length is 10 cm and height is 12 cm. Label the diagram with the measurements.
Applications of Pythagorean Theorem
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Slide 84 / 145
Find the base length of the pyramid whose height is 21 m and slant height is 29 m. Label the diagram with the measurements.
Applications of Pythagorean Theorem
Slide 84 (Answer) / 145
Find the base length of the pyramid whose height is 21 m and slant height is 29 m. Label the diagram with the measurements.
Applications of Pythagorean Theorem
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Ans
wer
Slide 85 / 145
40 The sizes of television and computer monitors are given in inches. However, these dimensions are actually the diagonal measure of the rectangular screens. Suppose a 14-inch computer monitor has an actual screen length of 11-inches. What is the height of the screen?
Slide 85 (Answer) / 145
40 The sizes of television and computer monitors are given in inches. However, these dimensions are actually the diagonal measure of the rectangular screens. Suppose a 14-inch computer monitor has an actual screen length of 11-inches. What is the height of the screen?
[This object is a pull tab]
Ans
wer x2 + 112 = 142
x2 + 121 = 196 x 2 = 75
Slide 86 / 145
41 Find the height of the pyramid whose base length is 16 in and slant height is 17 in. Label the diagram with the measurements.
Slide 86 (Answer) / 145
41 Find the height of the pyramid whose base length is 16 in and slant height is 17 in. Label the diagram with the measurements.
[This object is a pull tab]
Ans
wer
Slide 87 / 145
42 A tree was hit by lightning during a storm. The part of the tree still standing is 3 meters tall. The top of the tree is now resting 8 meters from the base of the tree, and is still partially attached to its trunk. Assume the ground is level. How tall was the tree originally?
Slide 87 (Answer) / 145
42 A tree was hit by lightning during a storm. The part of the tree still standing is 3 meters tall. The top of the tree is now resting 8 meters from the base of the tree, and is still partially attached to its trunk. Assume the ground is level. How tall was the tree originally?
[This object is a pull tab]
Ans
wer
32 + 82 = x2
9 + 64 = x2 73 = x2
The base of the tree is 3 m, the part that fell is 8.5 m tall, so the tree was a total of 11.5 m tall.
Slide 88 / 145
43 Suppose you have a ladder of length 13 feet. To make it sturdy enough to climb you myct place the ladder exactly 5 feet from the wall of a building. You need to post a banner on the building 10 feet above ground. Is the ladder long enough for you to reach the location you need to post the banner?Yes
No
Derived from( (
Slide 88 (Answer) / 145
43 Suppose you have a ladder of length 13 feet. To make it sturdy enough to climb you myct place the ladder exactly 5 feet from the wall of a building. You need to post a banner on the building 10 feet above ground. Is the ladder long enough for you to reach the location you need to post the banner?Yes
No
Derived from( (
[This object is a pull tab]A
nsw
er
Slide 89 / 145
44 You've just picked up a ground ball at 3rd base, and you see the other team's player running towards 1st base. How far do you have to throw the ball to get it from third base to first base, and throw the runner out? (A baseball diamond is a square)
home
1st
2nd
3rd
90 ft.
90 ft.90 ft.
90 ft.
Slide 89 (Answer) / 145
44 You've just picked up a ground ball at 3rd base, and you see the other team's player running towards 1st base. How far do you have to throw the ball to get it from third base to first base, and throw the runner out? (A baseball diamond is a square)
home
1st
2nd
3rd
90 ft.
90 ft.90 ft.
90 ft.
[This object is a pull tab]
Ans
wer 902 + 902 = x2
8100 + 8100 = x2 16,200 = x2
Slide 90 / 145
45 You're locked out of your house and the only open window is on the second floor, 25 feet above ground. There are bushes along the edge of your house, so you'll have to place a ladder 10 feet from the house. What length of ladder do you need to reach the window?
Slide 90 (Answer) / 145
45 You're locked out of your house and the only open window is on the second floor, 25 feet above ground. There are bushes along the edge of your house, so you'll have to place a ladder 10 feet from the house. What length of ladder do you need to reach the window?
[This object is a pull tab]
Ans
wer 102 + 252 = x2
100 + 625 = x2 725 = x2
26.9 feet = x
Slide 91 / 145
46 Scott wants to swim across a river that is 400 meters wide. He begins swimming perpendicular to the shore, but ends up 100 meters down the river because of the current. How far did he actually swim from his starting point?
Slide 91 (Answer) / 145
46 Scott wants to swim across a river that is 400 meters wide. He begins swimming perpendicular to the shore, but ends up 100 meters down the river because of the current. How far did he actually swim from his starting point?
[This object is a pull tab]
Ans
wer
4002 + 1002 = x2
160,000 + 10,000 = x2 170,000 = x2
400 m
100 m
Slide 92 / 145
Distance Formula
Click to return to the table of contents
Slide 93 / 145
5
10
5 10
-5
-5x
-10
0-10
y
If you have two points on a graph, such as (5,2) and (5,6), you can find the distance between them by simply counting units on the graph, since they lie in a vertical line.
The distance between these two points is 4.
The top point is 4 above the lower point.
Distance Between Two Points
Slide 94 / 145
5
10
5 10
-5
-5x
-10
0-10
y
47 What is the distance between these two points?
Slide 94 (Answer) / 145
5
10
5 10
-5
-5x
-10
0-10
y
47 What is the distance between these two points?
[This object is a pull tab]
Ans
wer The distance is 5.
The blue point is five to the right of the red point.Distance is always positive.
Slide 95 / 145
48 What is the distance between these two points?
5
10
5 10
-5
-5x
-10
0-10
y
Slide 95 (Answer) / 145
48 What is the distance between these two points?
5
10
5 10
-5
-5x
-10
0-10
y
[This object is a pull tab]
Ans
wer
3
Slide 96 / 145
5
10
5 10
-5
-5x
-10
0-10
y
49 What is the distance between these two points?
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Most sets of points do not lie in a vertical or horizontal line. For example:
Counting the units between these two points is impossible. So mathematicians have developed a formula using the Pythagorean theorem to find the distance between two points.
5
10
5 10
-5
-5x
-10
0-10
y
Distance Between Two Points
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5
10
5 10
-5
-5x
-10
0-10
y
Draw the right triangle around these two points. Then use the Pythagorean theorem to find the distance in red.
a
bc
Distance Between Two Points
Slide 98 (Answer) / 145
5
10
5 10
-5
-5x
-10
0-10
y
Draw the right triangle around these two points. Then use the Pythagorean theorem to find the distance in red.
a
bc
Distance Between Two Points
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Ans
wer
c2 = a2 + b2
c2 = 32 + 42
c2 = 9 + 16c2 = 25c = 5
The distance between the two points (2,2) and (5,6) is 5 units.
Slide 99 / 145
5
10
5 10
-5
-5x
-10
0-10
yExample:
Distance Between Two Points
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5
10
5 10
-5
-5x
-10
0-10
yExample:
Distance Between Two Points
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nsw
er
c2 = a2 + b2
c2 = 32 + 62
c2 = 9 + 36c2 = 45
The distance between the two points (-3,8) and (-9,5) is approximately 6.7 units.
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5
10
5 10
-5
-5x
-10
0-10
y
Distance Between Two PointsTry This:
Slide 100 (Answer) / 145
5
10
5 10
-5
-5x
-10
0-10
y
Distance Between Two PointsTry This:
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Ans
wer
c2 = a2 + b2
c2 = 92 + 122
c2 = 81 + 144c2 = 225c = 15
The distance between the two points (-5, 5) and (7, -4) is 15 units.
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Deriving a formula for calculating distance...
Distance Formula
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5
10
10
-5
-5x
-10
0-10
y
(x1, y1)
length = x2 - x1
length = y2 - y1
d
(x2, y2)
(x2, y1)
Create a right triangle around the two points. Label the points as shown. Then substitute into the Pythagorean Theorem.
c2 = a2 + b2
d2 = (x2 - x1)2 + (y2 - y1)2
d = (x2 - x1)2 + (y2 - y1)2
This is the distance formula now substitute in values.
Distance Formula
Slide 102 (Answer) / 145
5
10
10
-5
-5x
-10
0-10
y
(x1, y1)
length = x2 - x1
length = y2 - y1
d
(x2, y2)
(x2, y1)
Create a right triangle around the two points. Label the points as shown. Then substitute into the Pythagorean Theorem.
c2 = a2 + b2
d2 = (x2 - x1)2 + (y2 - y1)2
d = (x2 - x1)2 + (y2 - y1)2
This is the distance formula now substitute in values.
Distance Formula
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d = (5 - 2)2 + (6 - 2)2
d = (3)2 + (4)2
d = 9 + 16
d = 25
d = 5
Slide 103 / 145
Distance Formula
d = (x2 - x1)2 + (y2 - y1)2
You can find the distance d between any two points (x1, y1) and (x2, y2) using the formula below.
how far between the x-coordinate how far between the y-coordinate
Slide 104 / 145
When only given the two points, use the formula.
Find the distance between:Point 1 (-4, -7)Point 2 (-5, -2)
Distance Formula
Slide 104 (Answer) / 145
When only given the two points, use the formula.
Find the distance between:Point 1 (-4, -7)Point 2 (-5, -2)
Distance Formula
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Ans
wer
Slide 105 / 145
50 Find the distance between (2, 3) and (6, 8). Round answer to the nearest tenth.
Let:x1 = 2y1 = 3x2 = 6y2 = 8
hint
Slide 105 (Answer) / 145
Slide 106 / 145
51 Find the distance between (-7, -2) and (11, 3). Round answer to the nearest tenth.
Let:x1 = -7y1 = -2x2 = 11y2 = 3
hint
Slide 106 (Answer) / 145
Slide 107 / 145
52 Find the distance between (4, 6) and (1, 5). Round answer to the nearest tenth.
Slide 107 (Answer) / 145
Slide 108 / 145
53 Find the distance between (7, -5) and (9, -1). Round answer to the nearest tenth.
Slide 108 (Answer) / 145
53 Find the distance between (7, -5) and (9, -1). Round answer to the nearest tenth.
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Ans
wer
Slide 109 / 145
How would you find the perimeter of this rectangle?
Either just count the units or find the distance between the points from the ordered pairs.
Applications of the Distance Formula
Slide 109 (Answer) / 145
How would you find the perimeter of this rectangle?
Either just count the units or find the distance between the points from the ordered pairs.
Applications of the Distance Formula
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Ans
wer length = 8
width = 68 + 6 + 8 + 6 = 28
Slide 110 / 145
A (0,-1)B (8,0)
C (9,4)D (3,3)
Can we just count how many units long each line segment is in this quadrilateral to find the perimeter?
Applications of the Distance Formula
Slide 110 (Answer) / 145
A (0,-1)B (8,0)
C (9,4)D (3,3)
Can we just count how many units long each line segment is in this quadrilateral to find the perimeter?
Applications of the Distance Formula
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MP.1: Make sense of problems and persevere in solving them.
MP.2: Reasoning abstractly and quantitatively.MP.7: Look for and make use of structure.
Ask: What is the problem asking? (MP.1)How could you start this problem? (MP.1)How can you represent the problem with
symbols and numbers? (MP.2)How is finding the perimeter of a polygon in the coordinate plane related to the distance
formula? (MP.7)What do you know about the distance formula
that you can apply to this situation? (MP.7)
Slide 111 / 145
You can use the Distance Formula to solve geometry problems.
A (0,-1)B (8,0)
C (9,4)D (3,3)
Find the perimeter of ABCD.Use the distance formula to find all four of the side lengths.Then add then together.
BC =BC =
CD =CD =
AB =AB =
DA =DA =
Applications of the Distance Formula
Slide 111 (Answer) / 145
Slide 112 / 145
54 Find the perimeter of ΔEFG. Round the answer to the nearest tenth.
E (7,-1)
F (3,4)
G (1,1)
Slide 112 (Answer) / 145
Slide 113 / 145
55 Find the perimeter of the square. Round answer to the nearest tenth.
H (1,5)
I (3,3)K (-1,3)
J (1,1)
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55 Find the perimeter of the square. Round answer to the nearest tenth.
H (1,5)
I (3,3)K (-1,3)
J (1,1)[This object is a pull tab]
Ans
wer
Each side length is #8 So the perimeter is 4 times #8 # 11.3
Slide 114 / 145
56 Find the perimeter of the parallelogram. Round answer to the nearest tenth.
L (1,2) M (6,2)
N (5,-1)O (0,-1)
Slide 114 (Answer) / 145
Slide 115 / 145
Midpoints
Click to return to the table of contents
Slide 116 / 145
Find the midpoint of the line segment.
What is a midpoint?How did you find the midpoint?What are the coordinates of the midpoint?
Midpoint
5
10
5 10x0
y
(2, 2)
(2, 10)
-5
-5
-10
-10
Slide 117 / 145
Find the midpoint of the line segment.
What are the coordinates of the midpoint?
How is it related to the coordinates of the endpoints?
Midpoint
5
10
5 10x0
y
(3, 4) (9, 4)
-5
-10
Slide 117 (Answer) / 145
Find the midpoint of the line segment.
What are the coordinates of the midpoint?
How is it related to the coordinates of the endpoints?
Midpoint
5
10
5 10x0
y
(3, 4) (9, 4)
-5
-10
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Ans
wer
&
Mat
h Pr
actic
e
Midpoint = (6, 4)
It is in the middle of the segment.
Average of x-coordinates.Average of y-coordinates.
The questions on this slide address MP.7: Look for and make use of
structure.
Slide 118 / 145
The Midpoint Formula
To calculate the midpoint of a line segment with endpoints (x1,y1) and (x2,y2) use the formula:
( x1 + x2 y1 + y2
22, )
The x and y coordinates of the midpoint are the averages of the x and y coordinates of the endpoints, respectively.
Slide 119 / 145
The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B.
B (8,1)
A (2,5)
See next page for answer
The Midpoint Formula
Slide 120 / 145
The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B.
B (8,1)
A (2,5)
M
Use the midpoint formula:
The Midpoint Formula
( x1 + x2 y1 + y2
22, )
Slide 120 (Answer) / 145
The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B.
B (8,1)
A (2,5)
M
Use the midpoint formula:
The Midpoint Formula
( x1 + x2 y1 + y2
22, )
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Ans
wer
Substitute in values:2 + 8 , 5 + 1
2 2( )Simplify the numerators:
10 62 2
,
Write fractions in simplest form:( )
(5,3) is the midpoint of AB
Slide 121 / 145
Find the midpoint of (1, 0) and (-5, 3).
Use the midpoint formula:
The Midpoint Formula
( x1 + x2 y1 + y2
22, )
Slide 121 (Answer) / 145
Find the midpoint of (1, 0) and (-5, 3).
Use the midpoint formula:
The Midpoint Formula
( x1 + x2 y1 + y2
22, )
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Ans
wer
Substitute in values:1 + -5 , 0 + 3
2 2( )Simplify the numerators:
-4 32 2
,
Write fractions in simplest form:( )(-2,1.5) is the midpoint
Slide 122 / 145
57 What is the midpoint of the line segment that has the endpoints (2, 10) and (6 ,-4)?
A (3, 4)
B (4, 7)
C (4, 3)
D (1.5, 3)5
10
5 10
-5
-5x
-10
0-10
y
Slide 122 (Answer) / 145
57 What is the midpoint of the line segment that has the endpoints (2, 10) and (6 ,-4)?
A (3, 4)
B (4, 7)
C (4, 3)
D (1.5, 3)5
10
5 10
-5
-5x
-10
0-10
y
[This object is a pull tab]
Ans
wer
C
Slide 123 / 145
5
10
5 10
-5
-5x
-10
0-10
y
58 What is the midpoint of the line segment that has the endpoints (4, 5) and (-2, 6)?
A (3, 6.5)
B (1, 5.5)
C (-1, 5.5)
D (1, 0.5)
Slide 123 (Answer) / 145
5
10
5 10
-5
-5x
-10
0-10
y
58 What is the midpoint of the line segment that has the endpoints (4, 5) and (-2, 6)?
A (3, 6.5)
B (1, 5.5)
C (-1, 5.5)
D (1, 0.5)
[This object is a pull tab]
Ans
wer
B
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59 What is the midpoint of the line segment that has the endpoints (-4, -7) and (-12, 2)?
A (-8, -2.5)
B (-4, -4.5)
C (-1, -6.5)
D (-8, -4)
Slide 124 (Answer) / 145
59 What is the midpoint of the line segment that has the endpoints (-4, -7) and (-12, 2)?
A (-8, -2.5)
B (-4, -4.5)
C (-1, -6.5)
D (-8, -4)
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Ans
wer
A
Slide 125 / 145
60 What is the midpoint of the line segment that has the endpoints (10, 9) and (5, 3)?
A (6.5, 2)
B (6, 7.5)
C (7.5, 6)
D (15, 12)
Slide 125 (Answer) / 145
60 What is the midpoint of the line segment that has the endpoints (10, 9) and (5, 3)?
A (6.5, 2)
B (6, 7.5)
C (7.5, 6)
D (15, 12)
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Ans
wer C
Slide 126 / 145
61 Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2).
Which formula should be used to solve this problem?
A Pythagorean Formula
B Distance Formula
C Midpoint Formula
D Formula for Area of a Circle
Slide 126 (Answer) / 145
61 Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2).
Which formula should be used to solve this problem?
A Pythagorean Formula
B Distance Formula
C Midpoint Formula
D Formula for Area of a Circle[This object is a pull tab]
Ans
wer C
Since the center is at the midpoint of any diameter, find the midpoint of the two given endpoints.
Slide 127 / 145
62 Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2).
A (2.5,-2)B (2,2.5)C (-2,2.5)D (-1,1.5)
Slide 127 (Answer) / 145
62 Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2).
A (2.5,-2)B (2,2.5)C (-2,2.5)D (-1,1.5)
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Ans
wer
C
Slide 128 / 145
63 Find the center of the circle with a diameter having endpoints at (-12, 10) and (2, 6).
A (-7,8)B (-5,8)C (5,8)D (7,8)
Slide 128 (Answer) / 145
63 Find the center of the circle with a diameter having endpoints at (-12, 10) and (2, 6).
A (-7,8)B (-5,8)C (5,8)D (7,8)
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Ans
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B
Slide 129 / 145
If point M is the midpoint between the points P and Q. Find the coordinates of the missing point.
Use the midpoint formula and solve for the unknown.
M (8, 1)
P (8, -6)
Q = ?
Substitute
Multiply both sides by 2
Add or subtract
(8, 8)
( x1 + x2 y1 + y2
22, )
Using Midpoint to Find the Missing Endpoint
Slide 129 (Answer) / 145
If point M is the midpoint between the points P and Q. Find the coordinates of the missing point.
Use the midpoint formula and solve for the unknown.
M (8, 1)
P (8, -6)
Q = ?
Substitute
Multiply both sides by 2
Add or subtract
(8, 8)
( x1 + x2 y1 + y2
22, )
Using Midpoint to Find the Missing Endpoint
[This object is a pull tab]
Teac
her N
otes
Before moving onto the next slide, ask the class:
"Can you find a shortcut to solve this problem? How would your shortcut make the problem easier?"
The answer to this question is shown on the next slide.
Slide 130 / 145
If point M is the midpoint between the points P and Q. Find the coordinates of the missing point.
Another method that can be used to find the missing endpoint is to look at the relationship between both the x- and y-coordinates and use the relationship again to calculate the missing endpoint.
Following the pattern, we see that the coordinates for point Q are (8, 8), which is exactly the same answer that we found using the midpoint formula.
Using Midpoint to Find the Missing Endpoint
M (8, 1)
P (8, -6)
Q = ?
+7+0
+7+0
Slide 131 / 145
64 If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point?
A (-13, -22)B (-8.5, -9.5)C (-4.5, -7.5)D (-12.5, -6.5)
P = (-4,3)M = (-8.5,-9.5)Q = ?
Slide 131 (Answer) / 145
64 If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point?
A (-13, -22)B (-8.5, -9.5)C (-4.5, -7.5)D (-12.5, -6.5)
P = (-4,3)M = (-8.5,-9.5)Q = ?
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Ans
wer
A
Slide 132 / 145
65 If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point?
A (1, -1)B (-13, 19)C (-8, 11)D (-19, 8)
Q = (-6, 9)M = (-7, 10)P = ?
Slide 132 (Answer) / 145
65 If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point?
A (1, -1)B (-13, 19)C (-8, 11)D (-19, 8)
Q = (-6, 9)M = (-7, 10)P = ?
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Ans
wer
C
Slide 133 / 145
Glossary & Standards
Click to return to the table of contents
Slide 133 (Answer) / 145
Glossary & Standards
Click to return to the table of contents
[This object is a pull tab]
Teac
her N
otes Vocabulary Words are bolded
in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.
Slide 134 / 145
Back to
Instruction
Converse of Pythagorean TheoremIf a and b are measures of the shorter
sides of a triangle, c is the measure of the longest side, and c squared equals a
squared plus b squared, then the triangle is a right triangle.
3
4 542+32 = 52
16+9 = 2525 = 25
Example:
b
a c
a2+b2 = c2
right triangle
Slide 135 / 145
2 2 10 2
2 2 10 80 8
2 2 10 80 864 = 8
Formula: Distance
Back to
Instruction
DistanceLength
Measurement of how far two points are through space.
Slide 136 / 145
Back to
Instruction
HypotenuseThe longest side of a right
triangle that is opposite the right angle.
a2+b2 = c2
Slide 137 / 145
Back to
Instruction
Leg
2 sides that form the right angle of a right triangle.
a2+b2 = c2
Slide 138 / 145
Back to
Instruction
Midpoint
The middle of something. The point halfway along a line.
( x1 + x2 y1 + y2
22, )
Midpoint Formula:
( x1 + x2 y1 + y2
22, )
( 2 + 2 10 + 222
, )( 4 12
22, )
( 62 , )
Slide 139 / 145
Back to
Instruction
Postulate
Through any two points, there is exactly one line.
A property that is accepted without proof
AB
Segment Addition Postulate
CD
FE
CD + DE = CE
Angle Addition Postulate
G H
J
m∠FGJ + m∠JGH = m∠FGH
Slide 140 / 145
Back to
Instruction
Proof
Given: 3x - 24 = 0 Prove: x = 8
Reasoned, logical explanations that use definitions, algebraic properties, postulates,
and previously proven theorems to arrive at a conclusion
Statements Reasons1. 3x - 24 = 0 1. Given2. 3x = 24 2. Addition3. x = 8 3. Division
Slide 141 / 145
Back to
Instruction
Pythagorean TheoremIn a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of hypotenuse (c).
Formula:3
4 5
42+32 = 52
16+9 = 2525 = 25
Example:
Slide 142 / 145
Back to
Instruction
Pythagorean TriplesCombinations of whole
numbers that work in the Pythagorean Theorem.
3
4 5
42+32 = 52
16+9 = 2525 = 25
12
5 13
52+122 = 132
25+144 = 169169 = 169
4
5 7
52+42 = 72
25+16 = 4941 = 49
Slide 143 / 145
Back to
Instruction
Right Triangle
A triangle that has a right angle (90°).
sailstair case
30º
60º
45º
45º
Slide 144 / 145
Back to
Instruction
Two-Column Proof
Given: 3x - 24 = 0 Prove: x = 8
A tool to organize your reasoning into two columns. Statements are written in the left column. Reasons are written in the right
column.
Statements Reasons1. 3x - 24 = 0 1. Given2. 3x = 24 2. Addition3. x = 8 3. Division
Slide 145 / 145
Standards for Mathematical Practices
Click on each standard to bring you to an example of how to
meet this standard within the unit.
MP8 Look for and express regularity in repeated reasoning.
MP1 Make sense of problems and persevere in solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.