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NAME _____________________________________ COMMON CORE GEOMETRY

Unit 5

Setting up proportions with similar triangles

And Simplifying Radicals

DATE PAGE TOPIC HOMEWORK

12/19 2-3 Lesson 1: Similar triangles and proportions

Homework Worksheet

12/20 4-6 Lesson 2: Applying the Triangle Side Splitter Theorem

Homework Worksheet

12/21 7-8 Lesson 3: Ratios of Sides, Perimeters, and Areas

Homework Worksheet

12/22 Quiz No Homework

12/23-1/2 Christmas Break! Happy Holidays!

1/3 9-10 Review Lesson 4: The Angle Bisector Theorem

Homework Worksheet

1/4 11-13 Lesson 5: Special Relationships Within Right Triangles—Dividing into Two Similar Sub-Triangles

Homework Worksheet

1/5 14-15 Lesson 6: Special Relationships Within Right Triangles- Another useful proportion

Homework Worksheet

1/6 16-17 QUIZ LESSON 7: A side note- what to do if we get a radical?

No Homework

1/9 18-19 Lesson 8: More operations with radicals Homework Worksheet

1/10 20-21 LESSON 9: Adding and Subtracting Radicals Homework Worksheet

1/11 22-23 LESSON 10: Putting It all Together Review Worksheet

1/12 Review Ticket in

1/13 Test

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Lesson 1: Similar triangles and proportions

Prove triangles are similar:

a.) In the diagram below DE is parallel to AB, mark your picture accordingly:

b.) Fill in the appropriate givens -

Given: Prove: ∆𝐴𝐶𝐵~∆𝐷𝐶𝐸

c.) Proof:

d.) Draw the similar triangles separately and label:

e.) Now that we know that the triangles are similar, Let’s fill in the following proportions:

𝐴𝐶

𝐷𝐶=

𝐵𝐶

𝐶𝐷

𝐶𝐴=

𝐴𝐵

𝐷𝐸

=𝐶𝐵

𝐶𝐷

𝐶𝐸=

𝐶𝐴

A

D

C

E

B

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EXERCISES: In 1-3 find x given that 𝐷𝐸̅̅ ̅̅ ∥ 𝐴𝐶̅̅ ̅̅

1. 2.

3.

WORD PROBLEMS:

4. A vertical pole, 15 feet high, casts a shadow 12 feet long. At the same time, a nearby tree casts a shadow 40

feet long. What is the height of the tree?

5. Caterina’s boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feet

above the water in a lake. If she stands 10 feet from the edge of the cliff, she can visually align the top of the

cliff with the water at the back of her boat. Her eye level is 5.5 feet above the ground. How far out from the

cliff, to the nearest tenth, is Catarina’s boat?

35ft

10ft

5.5ft

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LESSON 2: Applying the Triangle Side Splitter Theorem

Side Splitter Theorem:

A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.

Using this Theorem, answer the following questions:

1. If 𝑋𝑌̅̅ ̅̅ ∥ 𝐴𝐶̅̅ ̅̅ , 𝐵𝑋̅̅ ̅̅ = 4, 𝐵𝐴̅̅ ̅̅ = 5, and 𝐵𝑌̅̅ ̅̅ = 6, what is 𝐵𝐶̅̅ ̅̅ ?

2. If 𝑋𝑌̅̅ ̅̅ ∥ 𝐴𝐶̅̅ ̅̅ , 𝐵𝑋̅̅ ̅̅ = 9, 𝐵𝐴̅̅ ̅̅ = 15, and 𝐵𝑌̅̅ ̅̅ = 15, what is 𝑌𝐶̅̅̅̅ ?

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6. A large flag pole stands outside of an office building. Josh realizes that when he looks up from the ground,

60m away from the flagpole, that the top of the flagpole and the top of the building line up. If the flagpole is

35m tall, and Josh is 170m from the building, how tall is the building to the nearest tenth?

BE CAREFUL!

The side splitter

theorem only works

for the sides that are

split NOT the parallel

bases!!!!

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Exercises

In exercises 1 and 2, find the value of x. Lines that appear to be parallel are in fact parallel

1.)

2.)

3.) In the diagram below, 𝐴𝐵̅̅ ̅̅ ∥ 𝐸𝐹̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ , AB=20, CD=8, FD=12 and AE:EC=1:3. If the perimeter of the

trapezoid ABCD is 64, find AE and EC.

THEOREM: If 3 or more lines are cut by 2 transversals, then the segments of the transversals are in

proportion.

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Lesson 3: Ratios of Sides, Perimeters, and Areas

Exercise

Given ABCA’B’C’ pictured to the right:

a. Find the lengths of the missing sides.

b. Find the perimeters of the triangles.

c. Find the areas of the triangles.

Area formula:

d. What is the ratio of the sides of the triangles (scale factor)?

e. What is the ratio of the perimeters of the triangles?

f. What is the ratio of the areas of the triangles?

RULES:

The ratio of the perimeters is _____________________________________________________

The ratio of the areas is _________________________________________________________

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Examples:

1.) When two figures are similar and the ratio of their sides is a:b, then:

The ratio of their perimeters is:

The ratio of their areas is:

2.) Two triangles are similar. The sides of the smaller triangle are 6,4,8. If the shortest side of the larger

triangle is 6, find the length of the longest side.

3.) The sides of a triangle are 8, 5, and 7. If the longest side of a similar triangle measures 24, find the

perimeter of the larger triangle.

4.) Find the ratio of the lengths of a pair of corresponding sides in two similar polygons if the ratio of the

areas is 4:25.

5.) The sides of a triangle are 7, 8 and 10. What is the length of the shortest side of a similar triangle whose perimeter is 75?

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Lesson 4: The Angle Bisector Theorem

The angle bisector theorem states:

Exercises:

1.) In ABC pictured below, AD is the angle bisector of A. If CD=6, CA=8 and AB=12, find BD.

2.) In ABC pictured below, AD is the angle bisector of A. If CD=9. CA=12 and AB=16. Find BD.

In ABC, if the angle bisector of A

meets side BC at point D, then

𝐵𝐷

𝐶𝐷=

𝐵𝐴

𝐶𝐴

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3.) The sides of ABC pictured below are 10.5, 16.5 and 9. An angle bisector meets the side length of 9. Find

the lengths of x and y.

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Lesson 5: Special Relationships Within Right Triangles—Dividing into Two Similar Sub-

Triangles

Opening Exercise

Use the diagram to complete parts (a)–(c).

a. Are the triangles shown similar? Explain.

b. Determine the unknown lengths of the triangles using Pythagorean Theorem.

Example 1

In ABC pictured to the right, B is a right angle and BC is the altitude.

DEFINE: Altitude: ____________________________________________________________________

a. How many triangles do you see in the figure? Draw them:

b. In △ 𝐴𝐵𝐶, the altitude 𝐵𝐷̅̅ ̅̅ divides the right triangle into two sub-triangles, ∆𝐵𝐷𝐶 and ∆𝐴𝐷𝐵.

Is ∆𝐴𝐵𝐶~ ∆𝐵𝐷𝐶?

Is ∆𝐴𝐵𝐶~ ∆𝐴𝐷𝐵?

Since ∆𝐴𝐵𝐶 ~ ∆𝐵𝐷𝐶 and △ 𝐴𝐵𝐶~ △ 𝐴𝐷𝐵, can we conclude that ∆𝐵𝐷𝐶~∆𝐴𝐷𝐵? Explain.

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Example 2

Consider the right triangle △ 𝐴𝐵𝐶 below.

a. Altitude 𝐵𝐷̅̅ ̅̅ is drawn from vertex 𝐵 to the line containing 𝐴𝐶̅̅ ̅̅ . Segment 𝐴𝐷̅̅ ̅̅ = 4 , segment 𝐷𝐶̅̅ ̅̅ = 𝑥,

and the segment 𝐵𝐷̅̅ ̅̅ as 8.

b. Draw the two similar triangles ADB and BDC and label the sides appropriately.

c. Set a proportion to find the value of 𝑥.

d. We should notice that the altitude of the whole triangle is the long leg in the small triangle and is the

short leg in the medium triangle. Therefore, we can really use a short cut when we see a diagram like

this:

=

4 x

8

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EXAMPLES: 1.) 2.)

3.) Given triangle 𝐼𝑀𝐽 with altitude 𝐽�̅�, find 𝐼𝐽, 𝐽𝐿, 𝑎𝑛𝑑 𝐽𝑀.

36

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Lesson 6: Special Relationships within Right Triangles another useful proportion

Consider the right triangle △ 𝐴𝐵𝐶 below (same as we worked with yesterday).

a. Altitude 𝐵𝐷̅̅ ̅̅ is drawn from vertex 𝐵 to the line containing 𝐴𝐶̅̅ ̅̅ . Segment 𝐴𝐷̅̅ ̅̅ = 4 , segment 𝐴𝐶̅̅ ̅̅ = 20,

and the segment 𝐵𝐴̅̅ ̅̅ as 𝑥.

b. Draw the two similar triangles ADB and ABC and label the sides appropriately.

c. Set a proportion to find the value of 𝑥.

d. We should notice that the hypotenuse of the small triangle is the leg in the large triangle and they

hypotenuse in the large triangle is the leg in the short triangle. Therefore, we can really use a short

cut when we see a diagram like this:

=

4

20

x

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EXAMPLES:

1.) 2.)

what is the length of ? What is the length of ?

3.) In the diagram below, the length of the legs and of right triangle ABC are 6 cm and 8 cm,

respectively. Altitude is drawn to the hypotenuse of . What is the length of to the nearest tenth of

a centimeter?

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LESSON 7: A side note- what to do if we get a radical? Opening Exercise Solve for x: Consider our answer, what can we do with it? If we round, we are not using the most accurate answer so we want to leave it in simplest radical form. (we’ll come back and do this later) Perfect Squares- ________________________________________________________________________ Identify some perfect squares below:

Factors: Perfect square

12

22

33

42

52

62

72

82

92

102

112

*make a note about this:

Factors: Perfect square

xx

x2x2

X3x3

X4x4

X5x5

X6x6

X7x7

6 8

x

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Write the square roots of the following:

1. √64 2. √𝑥8 3. √64𝑥8

4. √100𝑥4 5. √25𝑥6𝑦4

Simplify:

6. √32 7. √𝑥3 8. √32𝑥3

9. √75𝑥5 10. √72𝑥9𝑦4 11.) go back and simplify

your answer from the

opening exercise.

Simplifying Non-Perfect squares:

√𝑛𝑜𝑛 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑟𝑒 = √𝐿𝑎𝑟𝑔𝑒𝑠𝑡 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ √𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟

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Lesson 8: More operations with radicals

1. Multiplying radicals.

a. Compare the value of √36 to the value of

√9 × √4.

b. Compare the value of √144 to the value

of √16 × √9?

Multiplication of radicals is commutative- what does this mean in your own words?

2. Dividing Radicals

a. Compare the value of √100

25 to the value of

√100

√25.

b. Compare the value of √36

4 to the value of

√36

√4.

Division of radicals is commutative- what does this mean in your own words?

Exercises 3–12

Simplify each expression as much as possible.

3. √12 ∙ √3 =

4. √17

25=

CONCLUSION- RULES for Radicals:

Multiplication rule: √𝑎𝑏 = √𝑎 ∙ √𝑏 Division rule: √𝑎

𝑏=

√𝑎

√𝑏

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5. √5 ∙ √4 =

6. √36

√18=

7. √54 ∙ √2 =

8. √4

36=

Rationalize the Denominator:

If you end up with a radical in the denominator of your fraction you will need to rationalize it (get rid of

the radical on the denominator).

Examples:

a.) √1

8= b.) √

5

10 c.) √

9

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9. Find the area of the figure below: 10. Find the area of the triangle below:

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LESSON 9: Adding and Subtracting Radicals

Opening exercise:

1.)

a. Calculate the perimeter of the triangle below:

b. Calculate the perimeter of the triangle:

Since the radicals are not the same we need to do some work before we can add.

*Simplify each side then add.

2√2 3√18

√50

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SIMPLIFY

2.) 19√2 + 2√8 3.) 3√18 + 10√2

4.) 5√3 + √12. 5.) 3√500 + 2√125 − √45

Find the Perimeters

6.) 7.)

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LESSON 10: Putting It all Together

Opening Exercise

In the diagram below of right triangle ABC, an altitude is drawn to the hypotenuse .

Recall the proportions that we set up back in lessons 6 and 7 and fill in the appropriate proportions below:

𝒙

𝒛=

𝒄

𝒃=

𝒄

𝒂=

1.) Triangle ABC shown below is a right triangle with altitude drawn to the hypotenuse .

If and .

Write all answers in simplest radical form:

a.) what is the length of AD? C.) what is the length of AC?

b.) What is the length of AB? D.) What is the area of triangle ABC?

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2.) Given the diagram below, find:

a.) Length of AB

b.) Length of AD

c.) Length of AC

D.) Perimeter of triangle ABC.

E.) Express the area of triangle ABC in simplest form:

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