Name: ____________________________ 7.1 As the Crow Flies

Be the crow…

6. Use the distance formula to find the distance between the points P (4, -2) and T (-2, 3). Round

to the nearest tenth.

7. Find the distance between the points A (2, -6) and B (-4, -6).

How are point A and B alike?

8. Use the distance formula to find the length of CD and FE, leave your answers as a simplified

radicals. Then determine if 𝐶𝐷 ≅ 𝐹𝐸.

NOTES---NOTES- 7.2 Notes 7.2a Notes for partitioning a segment

To partition a segment we divide it into 2 pieces. The point at which we divide the segment is required to

follow a specified ratio.

Example: Below, the point P is positioned on 𝐴𝐵 so that P divides the segment in a 2:1 ratio. We would

say that the ratio of 𝐴𝑃: 𝑃𝐵 is 2:1

Our goal for today: Determine the location of a point P that divides a segment 𝐴𝐵 into the ratio a:b.

Example 1: Determine the coordinates of a point P that divides 𝐴𝐵 into such that 𝐴𝑃: 𝑃𝐵 is in a ratio of

1:2

Step 1: If the points A & B aren’t drawn in the plane, draw and connect

them.

Step 2: Convert the ratio 1:2 into an equivalent fraction.

Step 3: Determine the horizontal distance between A and B. Note whether

your direction from A to B is positive or negative in the x direction.

Step 4: Determine the vertical distance between A and B. Note whether the

direction from A to B is positive or negative in the y direction.

Step 5: To get the x coordinate of point P you must start at x coordinate of

A, and go 1/3rd

the horizontal distance from A to B. A formula for this is:

𝑋𝑃 = 𝑋𝐴 + 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 ∙ 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙

Step 6: To get the y coordinate of point P you must start at the y coordinate

of A, and go 1/3rd

the vertical distance from A to B. A formula for this is:

𝑌𝑃 = 𝑌𝐴 + 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 ∗ 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙

2 diagonal

lengths

1 diagonal

length

Example 2: Determine the location of point P that divides 𝐴𝐵 in the

ratio 𝐴𝑃: 𝑃𝐵 = 3: 2.

𝐴 = (−2, −2), 𝐵 = (4,1)

Example 3: Determine the location of point Q that divides 𝑋𝑌 in the ratio 𝑋𝑄: 𝑄𝑌 = 3: 5

𝑋 = (2,2), 𝑌 = (−5, −2)

𝑌𝑃

= 𝑌𝐴

+ 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛

∗ 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙

Step 1: If the points A & B aren’t drawn in the

plane, draw and connect them.

Step 2: Convert the ratio a:b into an equivalent

fraction.

Step 3: Determine the horizontal distance

between A and B. Note whether your direction

from A to B is positive or negative in the x

direction.

Step 4: Determine the vertical distance

between A and B. Note whether the direction

from A to B is positive or negative in the y

direction.

Step 5: To get the x coordinate of point P you

must start at x coordinate of A, and go the

fraction of the horizontal distance from A to

B. A formula for this is:

𝑋𝑃 = 𝑋𝐴 + 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 ∙ 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙

Step 6: To get the y coordinate of point P you

must start at the y coordinate of A, and go the

fraction of the vertical distance from A to B.

A formula for this is:

Name: _____________________________ Period: _________ 7.2b Partitioning Segments Homework

1. Determine the location of point P that divides 𝐴𝐵 in

the ratio 𝐴𝑃: 𝑃𝐵 = 2: 1. 𝐴 = (0,0), B= (4,4)

2. Determine the location of point P that divides 𝐴𝐵 in

the ratio 𝑃: 𝑃𝐵 = 3: 2. 𝐴 = (5,2), B= (0,4)

3. Determine the location of point T that divides 𝐺𝐻 in

the ratio 𝐺𝑇: 𝑇𝐻 = 4: 1. 𝐺 = (−4, −2), H= (5,2)

4. Determine the location of point C that divides 𝐴𝐵 in

the ratio 𝐴𝐶: 𝐶𝐵 = 1: 5. 𝐴 = (−4,0), B= (2,4)

5. Given M (5, -2) and N (-5, 3). Find the point P on 𝑀𝑁

such that P divides 𝑀𝑁 in the ratio 𝑀𝑃: 𝑃𝑁 1:3

6. Given J (-2, 5) and K (2, -3). Find the point P on

𝐽𝐾 such that P divides 𝐽𝐾 in the ratio 𝐾𝑃: 𝑃𝐽 3:5

7. The map shows a straight highway between two towns. Highway planners want to build two new rest

stops between the towns so that the two rest stops divide the highway into three equal parts. Find the

coordinates of the points at which the rest stops should be built. Please show your work.

8. Jennifer is thinking about purchasing a franchise of Taco Corp. Based on the geography of her town,

she wants to put her restaurant close to the school so the students being released from school will come

into her restaurant and buy tacos. About where would she put her restaurant if she wanted it ¼ the

distance between the school and the gas station, on the side closer to the school?

4

2

2

5

Bedford

Ashton

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Worksheet by Kuta Software LLC

Geometry

7.3 The equations of lines

Name___________________________________

Date________________ Period____

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-1-

Write the slope-intercept form of the equation of the line through the given points.

1) through: (-3, 0) and (-5, 5) 2) through: (1, -1) and (0, -5)

3) through: (3, 5) and (0, -2) 4) through: (-2, -3) and (-2, -5)

Write the slope-intercept form of the equation of the line through the given point with the givenslope.

5) through: (-5, 1), slope = 1

5

6) through: (-5, -3), slope = 0

7) through: (-2, -3), slope = 2

3

8) through: (4, 1), slope = 0

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Worksheet by Kuta Software LLC

-2-

Write the slope-intercept form of the equation of each line.

9)

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-2

2

4

10)

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-2

2

4

11)

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-2

2

4

12)

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-2

2

4

13) Find the equation of the line from the table ofvalues X Y-3 -8-1 -41 03 4

14) Find the equation of the line from the table ofvalues X Y-2 30 42 54 6

Name: ____________________________ Period: ________ 7.4 Write My Equation

Definitions:

Write the equation of the line, in slope-intercept form, as described below:

1. Triangle TRI has vertices: T(-3,-2), R(2,3), and I(5,0). Write the equation that contains the median of

the triangle that passes through vertex R.

2. Triangle NGL has the vertices: N(0,3), G(5,0), L(-4,-5). Write the equation of the line that contains

the altitude of the triangle that passes through vertex N.

Median: Extends from a

vertex to the midpoint of

the opposite side

Altitude: The perpendicular line which extends

from the line containing the base of a triangle to the

opposite vertex. It may lie outside the triangle.

Perpendicular Bisector:

The line that passes

through a segment’s

midpoint at a right angle

3. Triangle TAL has vertices: T(-7,-1), A(5,6), L(2,-3). Write the equation of the line that contains the

altitude of the triangle that passes through vertex T.

4. Write the equation of the perpendicular bisector of 𝐺𝑀̅̅̅̅̅ whose endpoints are G(-3,2), and M(7,4)

Name: _________________________ Period: ________ 7.5 Slope Criteria

Videos to help with this assignment if you were absent: https://www.youtube.com/watch?v=jIoj2oAR83k https://www.youtube.com/watch?v=X6uavZnHTuY

1. The points A (1,2) and B (3,1) are connected to form a segment. The points C (3,5) and D (5,4) are

connected to form a segment.

a) Are 𝐴𝐵 and 𝐶𝐷 parallel?

b) Would the segments 𝐴𝐶 and 𝐵𝐷 be parallel?

c) Demonstrate that it is possible to move point D so that 𝐴𝐵 ∥ 𝐶𝐷 but 𝐴𝐶 ∦ 𝐵𝐷. Explain why your

solution works.

2. The elevation data of a portion of the Appalachian Trail is shown.

a. Find the average slope between the 1st

and second marker

b. Find the average slope between marker

4 and marker 5.

c. What is the average slope between

marker 6 and the end of the trail?

3. Create your own contour profile. Each

contour line represents 20 feet of elevation

change. The scale is in feet.

a. Going from point A to point B create a contour profile in the graph below. Include units and a title.

b. How many total feet would you ascend on this journey?

c. What does it mean when contour lines are close together?

d. Where is the steepest section of this trip?

Name: ____________________________ Period: ________ 7.6 Properties of Triangles

SHOW ALL YOUR WORK. Provide the property that proves the triangles are congruent.

Distance formula:

1. Use the following list of points to determine if ∆𝑄𝑅𝑆 ≅ ∆𝑇𝑈𝑉

Q(-2,0), R(1,-2), S(-3,-2) T(5,1), U(3,-2), V(3,2)

2. Use the list of points to determine if ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹

A(2,3), B(3,-1), C(7,2) D(-3,1), E(1,2), F(-3,5)

SSS

SAS

ASA

AAS

HL

Use the given sets of points to prove each congruence statement.

3. F (-3, 3) A (-1, 3) N (-2, 0), H (0, -1) O (2, -1) T (1, 2)

prove: ∠FAN ≅ ∠HOT

4. W (2, 3) X (4, 1) A (1, -1), Z (-1, 0) G (-3, -2) O (0, -4)

prove: ∠WAX ≅ ∠ZOG

5. Are these diagrams correct or incorrect? If incorrect explain what is wrong.

Name: ____________________________ Period: ________ 7.7 Properties of Quadrilaterals

Determine whether each quadrilateral with the given vertices is a parallelogram, a trapezoid, or neither.

Graph. EXPLAIN your reasoning include calculations and words.

1) A (-9, -1) B (-5, 2) C (6, 2) D (-10, -10)

2) R (-8, 7) S (11, -8) T (2, 10) V (0,4)

3) J (4, 4) K (2, 1) L (-3, 2) M (-1, 5)

4) If a quadrilateral is a parallelogram with diagonals that are

perpendicular, then the figure is a rhombus. Determine whether

quadrilateral WXYZ is a rhombus. Explain in words and with

calculations.

5) Given 𝐴 = (0, −3), 𝐵 = (5,3), 𝑄 = (−3, −1) find two possible locations for a point P, such that 𝑃𝑄 ⃡

is parallel to 𝐴𝐵 ⃡ .

6) Quadrilateral OVER has vertices 𝑂(−4,2), 𝑉(1,1), 𝐸(0,6), 𝑅(−5,7).

a. Are the diagonals perpendicular? How do you know? Show your work

b. Find the midpoints of each diagonal. What can you conjecture about them?

c. Demonstrate what type of quadrilateral OVER is.

Name: _________________________________ Period: __________ 7.8 Circle Equations Day 1

Write the Equation of the circle with the given center and radius.

1. Center (0, 2); radius :5 2. Center (-1, 3); radius: 8

3. Center (-4,-5); radius: √2 4. Center (9, 0) radius: √3

Complete the square to put the equation of the circle into standard form, and then graph the circle.

5. 𝑥2 − 2𝑥 + 𝑦2 = 15 6. 𝑥2 + 4𝑥 + 𝑦2 − 6𝑦 = −9

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

7. Prove or disprove that the point (1, √3) lies on the circle that is centered at the origin and contains (0,2)

8. Prove or disprove that the point (2, √3) lies on the circle centered at the origin and contains the point (-3,0)

9. Prove or disprove via any method that the circle with equation 𝑥2 − 4𝑥 + 𝑦2 = −3 intersects the y-axis.

Name: ___________________________________Period: ________ 7.9 Deriving Circle Formula

1. Consider an arbitrary circle in the coordinate plane, with a center

located at the point (ℎ, 𝑘) and with a radius of length r.

A. Let P be a point on that circle with coordinates (𝑥, 𝑦) as in the

diagram.

Determine the coordinates of the point A: _____________

Determine the length CA = _________________________

Determine the length of PA = _______________________

Use the Pythagorean Theorem to write the relationship of the lengths of the right triangle ∆𝐶𝐴𝑃

_______________________ + _____________________ = _____________________

2. After comparing your work with your classmates, write the equation for a circle with a center (h,k) and

a radius of r.

3. If a circle’s center is at the origin, what is the equation for the circle in this case?

Write the Equation of the circle with the given center and radius.

4. Center (4,-2); radius 3 5. Center (-5, 8); radius: √7

Complete the square to put the equation of the circle into standard form, and then graph the circle.

6. 𝑥2 + 𝑦2 + 10𝑥 − 75 = 0 7. 𝑥2 + 6𝑥 + 𝑦2 − 4𝑦 = 15

8. Prove or disprove that the point (0, −3) lies on the circle that is centered at (-3, 2) and contains (3, 4)

Hint: what’s the radius?

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

Name: _________________________ Period: ________ 7.10 Review for Unit Exam

1. Given the points A(3, 4) and B(-6, -2)

a) Plot the two points, draw the segment 𝐴𝐵

b) Find the midpoint of 𝐴𝐵

c) Find the slope of 𝐴𝐵

d) Find equation of the line 𝐴𝐵

e) What is the slope of any line perpendicular to 𝐴𝐵

f) Write the equation of the perpendicular bisector 𝐴𝐵

2. Given the segment from C (0, -6) and D (6, 3)

a) What is slope of the line 𝐶𝐷?

b) What is the slope of any line parallel to 𝐶𝐷?

c) Find the length of 𝐶𝐷

(leave answer as simplified radical)

d) Find the point on 𝐶𝐷 that partitions 𝐶𝐷 into a ratio of 2:1

3. What is the center and radius of each circle?

a) (x – 9)2 + (y – 3)

2 = 1 center _________ radius __________

b) x2 + (y – 5)

2 = 100 center _________ radius __________

c) x2 + y

2 = 4 center _________ radius __________

4) Write the equation of the circle with the given information

a) use the graph to the right ____________________________________

b) Center (2, -4) radius 7 ____________________________________

5) Prove or disprove that the point (2, √2) is on the graph of the cirlce above (use your equation from 4a)

6) Triangle TVH has the vertices: T(0,5), V(5,0), H(-4,-5). Write the equation of the line that contains the

median of the triangle that passes through vertex T. (Be aware on the exam we may ask you to fine the

equation of a median, a perpendicular bisector, or other line. Please review those as well.)