geometry honors coordinate geometry · pdf filethe ratio is 2:1 but the segment has 3 parts....

1

GEOMETRY HONORS

COORDINATE

GEOMETRY PACKET

Name __________________________________

Period _________________________________

2

Day 1 - Directed Line Segments

DO NOW

Distance formula

2 2

1 2 1 2D x x y y

Midpoint formula

1 2 1 2,2 2

x x y yM

Slope formula

2 1

2 1

y ym

x x

Equation of a line

y mx b (Slope-Intercept Form)

𝑦 - 𝑦1 = 𝑚(𝑥 − 𝑥1) (Point-Slope Form)

1) What is the slope of the line passing through the

points 2,5 and 8,2 ? 2) If 3,4M is the midpoint of AB and the

coordinates of A are 9,8 , then find the coordinates

of B.

3) Graph the line that passes through 3, 2 and has

a slope of 2

3.

What is the equation of this line?

4) Given points C and D, find the distance of CD:

6,2 4, 8C D

3

A directed line segment is a segment that has distance and a direction. This definition basically defines a

segment to be a vector - magnitude and direction. The directed line segment AB implies that we are starting at

point A and going towards point B. This means that the initial point is A. This is important because when we

partition (or divide up) the segment into a ratio, the ratio will reference a starting point and a direction. For

example, partitioning directed line segment AB into a 1:3 ratio implies that we start at point A and then have 1

part followed by 3 parts for a total of 4 parts. A ratio of 3:1 would place the three parts first starting at point A

then then the 1 part would follow. This results in two very different partitions. The diagrams below

demonstrate these two different ratios.

Given the directed line segment AB , determine point P so that it divides the segment into the ratio of:

1:3 3:1

The ratio is 1:3 but the segment has 4 parts. The ratio is 3:1 but the segment has 4 parts.

We can reference the same partition of a line segment by using the different endpoints of the directed segment.

The diagram below demonstrates how you can reference the same location using either endpoint of the line

segment. Note how the wording changes for these two descriptions.

Directed line segment AB

is divided into a ratio of 1:2

Directed line segment BA

is divided into a ratio of 2:1

The ratio is 1:2 but the segment has 3 parts. The ratio is 2:1 but the segment has 3 parts.

Example 1: Determine the ratio of the directed line segment BA when partitioned by point P. (Hint: B is the initial point)

a) _____ : _____ b) _____ : _____ c) _____ : _____

3

1

P

A

B

3

1

P

A

B

2

1

P

A

B

2

1

P

A

B

P

A

B

P AB

P

A

B

4

Partitioning a Directed Line Segment Partitioning a line segment means to divide it up into pieces. To relate this to a dilation it means that we will be

doing a reduction (0 < k < 1) so that the point will be on the segment.

A is the initial point,

our center of dilation. 1

,2

( ) (4,2)A

D B 1,4

( ) (2,1)A

D B

Partitioning AB into a 1:1 ratio. Partitioning AB into a 1:3 ratio.

When we dilate by a value between 0 and 1 we place the image of our point onto the segment. This point

partitions the segment into two parts. Be careful, the scale factor is not exactly the partitioning ratio but they

definitely have a strong relationship. A scale factor represents the amount out of the total while the ratio is a

comparison of the two pieces. A little practice converts these two value back and forth easily.

Partition

Ratio

Dilation

Scale

Factor

Partition

Ratio

Dilation

Scale

Factor

Partition

Ratio

Dilation

Scale

Factor

Partition

Ratio

Dilation

Scale

Factor

1:2 2:1 3:4 4:5

If we move the initial point (the center of dilation) then we adjust our relationship to represent that change.

𝑫(𝒂,𝒃),𝒌 = (𝒌 ∙ ∆𝒙 + 𝒂, 𝒌 ∙ ∆𝒚 + 𝒃)

OR

Step 1: Find Horizontal Distance (run) = 𝒙𝟐 − 𝒙𝟏 = ∆𝒙

Step 2: Find Vertical Distance (rise) = 𝒚𝟐 − 𝒚𝟏 = ∆𝒚

Step 3: Find Dilation Scale Factor = 𝒌 = 𝒑𝒂𝒓𝒕

𝒘𝒉𝒐𝒍𝒆

Step 4: 𝑫(𝒂,𝒃),𝒌 = (𝒌 ∙ ∆𝒙 + 𝒂, 𝒌 ∙ ∆𝒚 + 𝒃)

A

B1

1

A

B

3

1

A

B

5

Ex. #2 - Determine the point P that partitions the directed line segment AB into a ratio of 1:1,

where A (-5,2) and B (3,6).

Ex. #3 - Determine the point P that partitions the directed line segment AB into a ratio of 2:3,

where A (1,-5) and B (9,-1).

6

Day 1 - HW

9

Day 2 - Writing Equations of a Line

Warm – Up

1. The dilation, 3

,4

( ) 'A

D B B partitions AB into a ratio AB’:B’B of:

A. 3:4 B. 1:3 C. 3:1 D. 3:7

2. Determine the ratio of the directed line segment AB when partitioned by point P. (Hint: A is the initial point)

a) _____ : _____ b) _____ : _____ c) _____ : _____

3. Given directed line segment AB , where A(0,0) and B(18,0). Determine the point P that partitions the

segment into a 5:1 ratio. A. P(3,0) B. P(6,0) C. P(12,0) D. P(15,0)

P

A

BP

A

BPA

B

10

Write an equation of a line that is parallel to the line 2x + y = 6 and whose y-intercept is the same as the line y = x - 2.

What do you know about slopes of parallel lines? _______________________

Method 1: Slope Intercept Form Method 2: Point Slope Form

Formula:

Formula:

1. Write an equation of a line perpendicular to y = 3

1x - 6 and has a y-intercept

equal to 10.

What do you know about slopes of perpendicular lines? _______________________


Formula:

Formula:

Answer: Answer:

Answer: Answer:

11

You try It! 3. Write an equation of a line that passes through the point (1, 5) and is

perpendicular to 2y = x - 6.

4. Write an equation of a line that is parallel to the line 6x – 2y = 14 with an x-intercept of 5.

Answer:

Answer:

12

Example 5: Write the equation of a line passing through the two points given.

(10, 20) and (20, 65)

Step 1: Calculate Slope -

Step 2: Use your calculate Slope to help you write your equation.


Formula:

Formula:

Practice: Write the equation of a line passing through the two points given.

(2, –5) and (–8, 5)

Answer: Answer:

Answer:

13

Horizontal and Vertical lines

Write the equation of the horizontal line and/or vertical passing through each point.

6. (3, 7)

7. (2, -4)

8.

9.

10. Write an equation of a line that is parallel to the y – axis and contains the point ( 6 , 1).

11. Write an equation of a line that is parallel to the x – axis and contains the point (7,

1

3).

Answer:

Horizontal line:

Vertical Line:

Answer:

Slope

Horizontal line:

Vertical Line:

Answer:

Answer:

14

SUMMARY

Memory Device for Vertical Lines Memory Device for Horizontal Lines

x = “a” value of point = (a, b) y = “b” value of point = (a, b)

Exit Ticket

1)

2)

15

Day 2 – Homework

16

8. Write (If possible, in point-slope form) an equation of the line…. a) Containing (2, 1) and (3, 4)

b) Containing (-6, 3) and (2, -1)

c) Containing (1, 5) and (-3, 5)

d) With an x-intercept of 2 and a slope of 7

e) That has an x-intercept of 3 and passes through (1, 8)

f) That passes through (-3, 6) and (-3, 10)

g) That passes through (8, 7) and is perpendicular to the graph of 3y = -2x + 24

17

9. The line that represents the equation y = 8x – 1 contains the point (k, 5). Find k.

11. Determine the point P that partitions AB , A(-6,8) B(1,-20) into a 3:4 ratio.

18

Day 3 - Writing Equations of Altitudes, Medians and Perpendicular Bisectors

Warm - Up

1.

2. Determine the point P that partitions AB , A(1,2) B(11,22) into a 1:4 ratio.

A. (3,6)

B. (5, 10)

C. (7, 14)

D. (9, 18)

19

A median of a triangle is a line segment drawn from the vertex of a triangle to the midpoint of the

opposite side.

“How to calculate the median 𝑨𝑫̅̅ ̅̅ to side 𝑩𝑪̅̅ ̅̅ of ABC”

A = (4, 10), B = (12, 6), and C = (8, 2)

Formula:

Formula:


Formula:

Formula:

Answer: Answer:

20

An altitude of a triangle is a line segment drawn from a vertex of a triangle perpendicular to the opposite

side.

“How to calculate the altitude 𝑨𝑫̅̅ ̅̅ to side 𝑩𝑪̅̅ ̅̅ of ABC”

A = (4, 10), B = (12, 6), and C = (8, 2)

Formula:

Formula:

Formula:


Formula:

Formula:

Answer: Answer:

21

A Perpendicular bisector of a line segment is a line (or line segment) that is perpendicular to the segment

at its midpoint.

“How to calculate the bisector 𝑨𝑫̅̅ ̅̅ to side 𝑩𝑪̅̅ ̅̅ of ABC”

A = (4, 10), B = (12, 6), and C = (8, 2)


Formula:

Formula:

Answer: Answer:

22

SUMMARY

Exit Ticket

23

Day 3 – Homework

In problems 13-17, use ∆ABC in the diagram.

13. Write, in point-slope form, an equation of a line through C parallel to 𝐴𝐵 ⃡ .

14. Write an equation of the perpendicular bisector of 𝐴𝐵̅̅ ̅̅ .

15. Write an equation of the altitude from C to 𝐴𝐵̅̅ ̅̅ .

16. Write an equation of the median form C to 𝐴𝐵̅̅ ̅̅ .

17. Find the slope of the line passing through the midpoints of 𝐴𝐶̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ .

24

18. A line passes through a point 3 units to the left of and 2 units above the origin.

Write an equation of the line if it is parallel to

a) The x-axis b) The y-axis

20. Determine the point P that partitions AB , A(-3,-14) B(21,2) into a 3:5 ratio.

25

Circles in the Coordinate Plane – Day 4

SWBAT: Write equations and graph circles in the coordinate plane.

Warm Up

Find the value that completes the square and then rewrite as a perfect square.

26

Example 1: Writing the Equation of a Circle

Model Problem: J with center J (2, 2) and radius 4

Practice #1: Write the equation of circle L with center L (–5, –6) and radius 9.

Example 2: Identifying the center and radius from the equation of a circle.

Model Problem:

Example 3: Identifying the center and radius from the graph of a circle.

Model Problem: Find the center and radius of the circle, and then write its equation.

27

Example 4: Writing the Equation of a Circle given a Center and a Point on the Circle

P with center P(0, –3) and passes through point (6, 5).

Example 5: Writing the Equation of a Circle given Two Endpoints.

Model Problem: Writing the equation of K that passes through endpoints A(5, 4) and B(1, –8).

Step 1: Calculate the Midpoint.

Step 2: Calculate the radius of the circle.

Step 3: Plug in Midpoint, and radius into circle equation.

Practice #2: Write the equation of circle Q that passes through (2, 3) and (2, –1)

28

Example 6: Write the equation of the circle from General Form to Standard Form.

Write the equation of the circle and identify the center and radius.

Practice #3: Write the equation of the circle and identify the center and radius.

29

Example 7: Graphing Circles Practice #6

Graph: x2 + y

2 = 16. Graph: (x +5)

2 + (y - 2)

2 = 4.

Challenge

30

SUMMARY

Exit Ticket

1. 2.

31

Homework – Day 4

33

A circle passes through 2 points (2, 2) and

(4, 2). Write the equation of the circle.

34

7)

8)

9)

35

Day 5: System of Equations SWBAT: Graph the Solutions to Quadratic Linear Systems

Warm – Up

Algebra

36

Example:

2.

37

Quadratic Linear System of Equations

Quadratic Equation

Vertex = ( , )

Linear Equation

y = x + 3

m = _____

b = ____

38

Quadratic Equation

Vertex = ( , )

Linear Equation

m = _____

b = ____

2.

39

3.

4.

40

Summary

Exit Ticket

41

Day 5 – HW

1.

2.

Graph the system and find the points of intersection.

3.

42

4.

5.

43

6.

7.

44

Day 6: Writing Equations of a Line Using Points of Intersection

1. Write the equation of a line that contains the point of intersection of the graphs

x = 4 and y = 2x + 8 and is parallel to the line whose equation is y = -2x + 5.

Step 1: Solve the system of equations to determine the point of intersection.

Step 2: Write the equation of your line using the point of intersection from above.


Formula:

Formula:

Answer: Answer:

45

2. Write the equation of a line that contains the point of intersection of the graphs 8x – 3y = 7 and

10x + 4y = -1 and is perpendicular to the line .73

1 xy

Step 1: Solve the system of equations to determine the point of intersection.

Step 2: Write the equation of your line using the point of intersection from above.


Formula:

Formula:

Answer: Answer:

46

Using Equations of Lines to Find the Coordinates of an Altitude

3. In ABC with coordinates A(-3,4), B(6,-2) and C(7,6) altitude 𝐶𝐷̅̅ ̅̅ is drawn.

Find the coordinates of D.

Step 1: Calculate the equation of the line where the altitude intersects the side.

Step 2: Calculate the equation of the altitude.

Step 3: Solve the systems from above for D

47

You try!

a) In ABC with coordinates A(0,0), B(6,3) and C(1,5) altitude CD is drawn. Find the

coordinates of D.

Ans: D = ( 𝟏𝟒

𝟓,

𝟕

𝟓)

48

b) In ABC with coordinates A

2

14,4,

2

1,4 B , and C

2

13,4 altitude BD is drawn, find

the coordinates of D.

Ans: D = (-2.6849, 0.9931)

49

c) In ABD with coordinates A(-4,1), B(1,5) and C(6, -1) altitude CD is drawn. Find the

coordinates of D.

Ans: D = (1.1219, 5.0976)

50

Day 6 - Homework

7) Show that the graphs of the following 3 equations are concurrent (intersect at a single point).

What are the coordinates of the point of intersections?

{

2𝑥 + 3𝑦 = 2𝑦 = 2𝑥 − 103𝑥 − 𝑦 = 14

9) Find, in point-slope form, an equation of the line containing (2, 1) and the point of intersection

of the graphs of 3x – y = 3 and x + 2y = 15.

51

10) Find an equation of the line that is parallel to the graph of 2x + 3y = 5 and contains the point of

intersection of the graphs of y = 4x + 8 and y = x + 5.

52

13) In ABD with coordinates A(-4,1), B(1,5) and C(6, -1) altitude CD is drawn. Find the

coordinates of D.

53

15) Find the distance between the parallel lines corresponding to y = 2x + 3 and y = 2x + 7.

(Hint: Start by choosing a convenient point on one of the lines.)

55

4. The vertices of triangle PQR are P(1,2), Q(-3,6) and R(4, 8) .

a. Find the coordinates of S, the midpoint of 𝑃𝑄̅̅ ̅̅ .

b. Express in radical form, the length of the median 𝑅𝑆̅̅̅̅ .

c. Find the slope of 𝑃𝑅̅̅ ̅̅ .

d. A line through point Q is parallel to 𝑃𝑅̅̅ ̅̅ . If the line passes through the point (x, 14), find the

value of x.

5. Find the coordinates of the midpoint of CD: C

2

1,

2

231 D

3

32,

3

25

6. Find the distance from S to T: S (x + y, a + b) T (x – y, b - a)

7. Simplify: a) 503 b) 122 c) 67153

56

8. In ABD with coordinates A(-4,1), B(1,5), C(6, -1) altitude CD is drawn. Find the coordinates of D.

9. Write an equation of the perpendicular bisector of the segment that joins the points (3, -7 ) and (5,1).

10. Write an equation of a line that passes through the point B(3,1) and is perpendicular to the line 3y + 2x = 15.

57

11. The vertices of ABC are A(0,6), B( -8, 0), C(0,0). Write an equation of the line that passes through one of

the vertices of the triangle and parallel to 𝐴𝐶̅̅ ̅̅ .

12. Write an equation of the line that contains point (-5,2) and is parallel to the y – axis.

13. Write an equation of the line that contains point (4,-1) and is perpendicular to the x – axis.

14. Write an equation of a line that contains the point (2, 2) and the intersection of the graphs x + y = 10 and

x –y = 2.

58

15. In ABC with coordinates A(-4,3), B(2,7) and C(4,-3).

a) Find the equation of the median to 𝐴𝐶̅̅ ̅̅ .

b) Find the equation of the altitude to 𝐴𝐵̅̅ ̅̅ .

c) Find the equation of the perpendicular bisector of 𝐴𝐵̅̅ ̅̅ .

59

16. The dilation, 3

,4

( ) 'A

D B B partitions AB into a ratio AB’:B’B of:

A. 3:4 B. 1:3 C. 3:1 D. 3:7

17. Directed line segment AB is partitioned by point P into a ratio of 4:1. Which of the following

represent this relationship?

A.

B.

C.

D.

18. Determine the point P that partitions AB , A(-3,-14) B(21,2) into a 3:5 ratio.

PB A PA B

PA B PA B

60

19. Which of the following is true about the equation: x2 + (y - 8)2 = 3? a) The center is at the origin and the radius is 3. b) The center is at (0,8) and the radius is 3. c) The center is at (0,8) and the radius is 9.

d) The center is at (0,8) and the radius is 3 .

20. Given the equation: x2 + y2 + 6x - 6y + 6 = 0 a) Write each equation in center-radius form. b) Find the coordinates of the center. c) Find the radius of the circle. d) Graph the circle.

21. Write the equation of a circle given center (6, 5) and the point on the circle (0, -3). Then graph.

61

22.

Part a: Find the points of intersection of the line y = 2x + 1 and the circle x2 + y

2 - 2y – 4 = 0

Part b: Show that the line y = 2x + 1 is a diameter of the circle.

62

Answer Key:

1. (-16,16)

2. a) b) 2

226

3. k = -1

4. a) (-1,4) b) 41 c) 2 d) x =1

5.

12

327,

12

2713

6. 2222 244 ayay

7. a) 15 2 b) 4 3 c. 63 10

8. equation of 𝐴𝐵̅̅ ̅̅ : y – 1 = 4

5(𝑥 + 4); y

5

21

5

4x

equation of altitude: y + 1 = - 5

4(𝑥 − 6); y =

2

13

4

5 x

coordinates of D:

41

209,

41

46

9. y + 3 = - 1

4(𝑥 − 4) 𝑜𝑟 y = -

1

4𝑥 - 2

10. y – 1 = 3

2(𝑥 − 3) 𝑜𝑟 y =

3

2𝑥 –

7

2

11. x = -8

12. x = -5

13. x = 4

14. y – 2 = 1

2(𝑥 − 2) 𝑜𝑟 y – 4 =

1

2(𝑥 − 6) 𝑜𝑟 y =

1

2𝑥 + 1

15. a) y – 0 = 7

2(𝑥 − 0) 𝑜𝑟 y =

7

2𝑥

b) y + 3 = −3

2(𝑥 − 4) 𝑜𝑟 y = -

3

2𝑥 + 3

c) y - 5 = −3

2(𝑥 + 1) 𝑜𝑟 y = -

3

2𝑥 +

7

2

16. C

17. D

18. (6,-8)

-5

3

63

19. D

20.

21.

22.

64

Additional Questions: Honors Text Book – Pages: 644 – 647: 13, 17, 21ab, 29

13)

17)

21a)

21b)

𝑜𝑟 y – 4 = 2(𝑥 − 7)

geometry honors coordinate geometry · pdf filethe ratio is 2:1 but the segment has 3 parts....

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