Analytic Geometry - content.njctl.orgcontent.njctl.org › courses › math › geometry-2015-16 › ... · Geometry – Analytic Geometry ~2~ NJCTL.org Midpoint Formula: CLASSWORK

Geometry – Analytic Geometry ~1~ NJCTL.org

Analytic Geometry

Distance Formula: CLASSWORK

1. Find the distance between (-9, 1) & (-5, -2)

2. Find the distance between (10, 3) & (1, -3)

3. Length of 𝐵𝐷̅̅ ̅̅ = ?

4. Length of 𝐴𝐷̅̅ ̅̅ = ?

5. Length of 𝐷𝐶̅̅ ̅̅ = ?

Distance Formula: HOMEWORK

6. Find the distance between (2, 9) & (-3, 14)

7. Find the distance between (-3, 2) & (9, 7)

8. length of 𝐴𝐷̅̅ ̅̅ = ?

9. length of 𝐵𝐷̅̅ ̅̅ = ?

10. length of 𝐶𝐷̅̅ ̅̅ = ?

#3 - 5

#8 - 10

Geometry – Analytic Geometry ~2~ NJCTL.org

Midpoint Formula: CLASSWORK

Calculate the coordinates of the midpoint of the given segments

11. (0, 0), (6, 10)

12. (2, 3), (6, 7)

13. (4, -1), (-2, 5)

14. (-3, 8), (13, -6)

15. (-1, -14), (-2, -6)

16. (3, 2), (6, 6)

17. (-5, 2), (0, 4)

Calculate the coordinates of the other endpoint of the segment with the given endpoint and midpoint M

18. endpoint: (4,6), midpoint: (7,11)

19. endpoint: (2, 6), midpoint:

(-1, 1)

20. endpoint: (3, -12), midpoint (2,-

1)

Midpoint Formula: HOMEWORK

Calculate the coordinates of the midpoint of the given segments

21. (0, 0), (8, 4)

22. (-1, 3), (7, -1)

23. (3, 5), (7, -9)

24. (6, 0), (2, 7)

25. (-5, -3), (-3, -5)

26. (13, 8), (-6, -6)

27. (-4, -2), (1, 3)

Calculate the coordinates of the other endpoint of the segment with the given endpoint and midpoint M

28. endpoint: (-5, 9)

midpoint (-8, -2)

29. endpoint: (6, 7)

midpoint (10, -7)

30. endpoint: (2, 4)

midpoint (-1, 7)

3

Partitions of a Line Segment: CLASSWORK

PARCC-type Questions:

31. Line segment AB in the coordinate plane has endpoints with coordinates A (3, -10) and B(-6, -1).

a) Graph 𝐴𝐵̅̅ ̅̅

b) Find 2 possible locations for point C so that C divides 𝐴𝐵̅̅ ̅̅ into 2 parts with lengths in a ratio of 1:2.

32. Line segment EF in the coordinate plane has endpoints with coordinates E (-10, 11) and F (5, -9).

a) Graph 𝐸𝐹̅̅ ̅̅

b) Find 2 possible locations for point G so that G divides 𝐸𝐹̅̅ ̅̅ into 2 parts with lengths in a ratio of 2:3.

4

33. Line segment JK in the coordinate plane has endpoints with coordinates J (11, 11) and K (-10, -10). Find

two possible locations for point P that divides 𝐽𝐾̅̅ ̅ into two parts with lengths in a ratio of 3:4.

34. Line segment LM in the coordinate plane has endpoints with coordinates L (-12, 10) and M (6, -8). Find

two possible locations for point P that divides 𝐿𝑀̅̅ ̅̅ into two parts with lengths in a ratio of 2:7.

35. Line segment RS in the coordinate plane has endpoints with coordinates R(7, -11) and S(-9, 13). Find two

possible locations for point P that divides 𝑅𝑆̅̅̅̅ into two parts with lengths in a ratio of 3:5.

Partitions of a Line Segment: HOMEWORK

PARCC-type Questions:

36. Line segment AB in the coordinate plane has endpoints with coordinates A (5, -7) and B(-10, 3).


b) Find 2 possible locations for point C so that C divides 𝐴𝐵̅̅ ̅̅ into 2 parts with lengths in a ratio of 1:4.

5

37. Line segment EF in the coordinate plane has endpoints with coordinates E (-10, 11) and F (10, -9).

a) Graph 𝐸𝐹̅̅ ̅̅

b) Find 2 possible locations for point G so that G divides 𝐸𝐹̅̅ ̅̅ into 2 parts with lengths in a ratio of 7:3.

38. Line segment JK in the coordinate plane has endpoints with coordinates J (11, 11) and K (-10, -10). Find

two possible locations for point P that divides 𝐽𝐾̅̅ ̅ into two parts with lengths in a ratio of 2:5.

39. Line segment LM in the coordinate plane has endpoints with coordinates L (-12, 10) and M (6, -8). Find

two possible locations for point P that divides 𝐿𝑀̅̅ ̅̅ into two parts with lengths in a ratio of 5:4.

40. Line segment RS in the coordinate plane has endpoints with coordinates R(7, -11) and S(-9, 13). Find two

possible locations for point P that divides 𝑅𝑆̅̅̅̅ into two parts with lengths in a ratio of 1:7.

6

Slopes of Parallel & Perpendicular Lines: CLASSWORK

Identify the slope of the line containing the given points:

41. (-2,11), (-5, 2)

42. (-4,11), (-4, -7)

43. (-2,22), (-5, 22)

44. Is the following system of equations parallel? Justify your answer.

A B

45. Is the following system of equations perpendicular? Justify your answer.

A B

46. If one line has a slope of -5, what must be the slope of a line parallel to it?

47. If one line passes through the points (-1, 2) & (7, 6), what must be the slope of a line parallel

to it?

48. If one line passes through the points (3, -5) & (1, 9) and a parallel line passes through the

point (3, 4), what is the other point that would lie on the 2nd line?

7

49. If one line has a slope of ½, what must be the slope of a line perpendicular to it?

50. If one line passes through the points (3, -5) & (1, 9), what must be the slope of a line

perpendicular to it?

51. If one line passes through the points (5, 2) & (7, 6) and a perpendicular line passes through

the point (3, 4), what is the other point that would lie on the 2nd line?

Slopes of Parallel & Perpendicular Lines: HOMEWORK

Identify the slope of the line containing the given points:

52. (-6,12), (-2, 5)

53. (14,11), (-14, 11)

54. (-2,17), (-2, 18)

55. Is the following system of equations parallel? Justify your answer.

A B

56. Is the following system of equations perpendicular? Justify your answer.

A B

8

57. If one line has a slope of ½, what must be the slope of a line parallel to it?

58. If one line passes through the points (3, -5) & (1, 9), what must be the slope of a line parallel

to it?

59. If one line passes through the points (5, 2) & (7, 6) and a parallel line passes through the

point (3, 4), what is another point that would lie on the 2nd line?

60. If one line has a slope of -5, what must be the slope of a line perpendicular to it?

61. If one line passes through the points (-1, 2) & (7, 6), what must be the slope of a line

perpendicular to it?

62. If one line passes through the points (3, -5) & (1, 9) and a perpendicular line passes through

the point (3, 4), what is another point that would lie on the 2nd line?

Equations of Parallel & Perpendicular Lines: CLASSWORK

63. Find an equation of the line in point-slope form passing through point (-2, 5) and parallel to

the line whose equation is 4x – 2y = -5

64. Two lines are represented by equations: 2x + 4y = 21 and y = kx – 12. What value of k will

make lines parallel?

65. Find an equation of the line in slope-intercept form passing through point (-4,6) and parallel

to the line whose equation is y = -¾ x + 11

66. The sides of a quadrilateral lie on the lines y = 4x + 5, y = 1/3x +7, 8x – 2y = 1, and

x – 3y = 2. Is the quadrilateral a parallelogram? Justify your answer.

67. Find an equation of the line passing through point (4, -5) and perpendicular to the line whose

equation is 3x – 6y = -11.

68. Two lines are represented by equations: -3x + 6y =21 and y = kx +5. What value of k will

make lines perpendicular?

69. Find an equation of the line passing through point (8, -2) and perpendicular to the line whose

equation is y = 4x + 11.

70. The sides of a quadrilateral lie on the lines y= 4x + 5, y = 1/3x + 7, x + 4y = 1, and

x – 3y = 2, is the quadrilateral a rectangle? Justify your answer.

71. Determine if the following equations are parallel, perpendicular, or neither. Justify your

answer. 4x + 3y = 9 6x – 8y = 20

9


answer. 5(x + 3) = 3y + 12 5x + 3y = 15

Equations of Parallel & Perpendicular Lines: HOMEWORK

73. Find an equation of the line in point-slope form passing through point (-3, 2) and parallel to

the line whose equation is 6x – 2y = 7.

74. Two lines are represented by equations: -4x + 12y = 21 and y = kx – 12.

What value of k will make lines parallel?

75. Find an equation of the line in point-slope form passing through point (-8,3) and parallel to

the line whose equation is y = -¾ x + 11.

76. The sides of a quadrilateral lie on the lines 3x + y = 7, x + y = 12, 6x – 2y = 2, and x – y = 2.

Is the quadrilateral a parallelogram? Justify your answer.

77. Find an equation of the line passing through point (-6,2) and perpendicular to the line whose

equation is 4x + 6y = -1

78. Two lines are represented by equations: 10x – 15y = 21 and y = kx + 5.

What value of k will make lines perpendicular?

79. Find an equation of the line passing through point (8,-2) and perpendicular to the line whose

equation is y = -2x + 11

80. The sides of a quadrilateral lie on the lines 4x – y = 5, x + 4y = 7, 8x – 2y = 1, and

3x + 12y = 2. Is the quadrilateral a rectangle? Justify your answer.


answer. 9x + 5y = 32 4.5x + 2.5y = 7.5


answer. 7(x – 1) = 3y + 21 3.5x + 1.5y = 4.5

10

Triangle Coordinate Proofs: CLASSWORK – CP

For numbers 83 – 84 find (a.) the length of and (b.) the midpoint coordinates of .

83. A(6, 7), B(4, 3) 84. A(1, 5), B(2, 3)

85. If one line has a slope of -3 what must be the slope of (a.) any line parallel to it and (b.) any

line perpendicular to it?

86. If one line passes through the points (0,3) and (-4,1) what must be the slope of (a.) any line

parallel to that first line and (b.) any line perpendicular to that first line?

87. Given: 𝐺𝐽⃗⃗⃗⃗ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑂𝐺𝐻

Prove: ∆𝐺𝐽𝑂 ≅ ∆𝐺𝐽𝐻

88. Given: Coordinates of the vertices of ∆𝑂𝑃𝑀 & ∆𝑂𝑁𝑀

Prove: ∆𝑂𝑃𝑀 & ∆𝑂𝑁𝑀 are congruent

isosceles triangles

AB AB

Statements Reasons

1. 𝐺𝐽⃗⃗⃗⃗ bisects OGH 1.

2. 2. Definition of an

angle bisector

3. 𝐺𝐽̅̅ ̅ ≅ 𝐺𝐽̅̅ ̅ 3.

4. 𝑂𝐺 = ___________, 𝐻𝐺 = ___________

4. Distance Formula

5. 𝑂𝐺̅̅ ̅̅ ≅ 𝐻𝐺̅̅ ̅̅ 5.

6. GJO GJH 6.

Statements Reasons

1. Coordinates of

vertices of OPM

and ONM

1.

2. OP = ______,

PM = ______,

MN = ______,

NO = ______

2. Distance Formula

3. 𝑂𝑃̅̅ ̅̅ ≅ 𝑃𝑀̅̅̅̅̅ ≅𝑀𝑁̅̅ ̅̅ ̅ ≅ 𝑁𝑂̅̅ ̅̅

3.

4. 𝑂𝑀̅̅ ̅̅ ̅ ≅ 𝑂𝑀̅̅ ̅̅ ̅ 4.

5. OPM and

ONM are isosceles

5.

6. OPM ONM 6.

11

Triangle Coordinate Proofs: HOMEWORK – CP


89. A(14, 2), B(7, 8) 90. A(0, 0), B(5, 12)

91. If one line has a slope of 2/7 what must be the slope of (a.) any line parallel to it and (b.) any


92. If one line passes through the points (-2,5) and (7,-1) what must be the slope of (a.) any line


93. Given: 𝑂𝑆⃗⃗⃗⃗ ⃗ ⊥ 𝑅𝑇̅̅ ̅̅

Prove: 𝑂𝑆̅̅̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑇𝑂𝑅

94. Given: G is the midpoint of 𝐻𝐹⃗⃗ ⃗⃗ ⃗ Prove: ∆𝐺𝐻𝐽 ≅ ∆𝐺𝐹𝑂

AB AB

Statements Reasons

1. 1. Given

2. RSO and TSO

are right angles

2.

3. RSO ≅ TSO 3.

4. 𝑂𝑆̅̅̅̅ ≅ 𝑂𝑆̅̅̅̅ 4.

5. OR = 6, OT = 6 5.

6. 𝑂𝑅̅̅ ̅̅ ≅ 𝑂𝑇̅̅ ̅̅ 6.

7. SOR SOT 7.

8. SOR SOT 8.

9. 𝑂𝑆⃗⃗⃗⃗ ⃗ bisects TOR 9.

Statements Reasons

1.

1. Given

2. 2. Vertical Angles

are

3. 𝐻𝐺̅̅ ̅̅ ≅ 𝐹𝐺̅̅ ̅̅ 3.

3. 𝑂𝐺 = ___________ 𝐽𝐺 = __________

4. Distance Formula

5. 𝑂𝐺̅̅ ̅̅ ≅ 𝐽𝐺̅̅ ̅ 5.

6. ∆𝐺𝐻𝐽 ≅ ∆𝐺𝐹𝑂 6.

G

12

Triangle Coordinate Proofs: CLASSWORK – Honors


83. A(6, 7), B(4, 3) 84. A(1, 5), B(2, 3)

85. If one line has a slope of -3 what must be the slope of (a.) any line parallel to it and (b.) any


86. If one line passes through the points (0,3) and (-4,1) what must be the slope of (a.) any line


87. Given: 𝐺𝐽⃗⃗⃗⃗ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑂𝐺𝐻

Prove: ∆𝐺𝐽𝑂 ≅ ∆𝐺𝐽𝐻

88. Given: Coordinates of the vertices of

∆𝑂𝑃𝑀 & ∆𝑂𝑁𝑀

Prove: ∆𝑂𝑃𝑀 & ∆𝑂𝑁𝑀 are congruent isosceles triangles

AB AB

13

Triangle Coordinate Proofs: HOMEWORK – Honors


89. A(14, 2), B(7, 8) 90. A(0, 0), B(5, 12)

91. If one line has a slope of 2/7 what must be the slope of (a.) any line parallel to it and (b.) any


92. If one line passes through the points (-2,5) and (7,-1) what must be the slope of (a.) any line


93. Given: 𝑂𝑆⃗⃗⃗⃗ ⃗ ⊥ 𝑅𝑇̅̅ ̅̅

Prove: 𝑂𝑆̅̅̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑇𝑂𝑅

94. Given: G is the midpoint of 𝐻𝐹⃗⃗ ⃗⃗ ⃗ Prove: ∆𝐺𝐻𝐽 ≅ ∆𝐺𝐹𝑂

AB AB

G

14

Equations of a Circle & Completing the Square: CLASSWORK

What are the center and the radius of the following circles?

95. (𝑥 + 2)2 + (𝑦 − 4)2 = 16

96. (𝑥 − 3)2 + (𝑦 − 7)2 = 25

97. (𝑥)2 + (𝑦 + 8)2 = 1

98. (𝑥 − 7)2 + (𝑦 + 1)2 = 17

99. (𝑥 + 6)2 + (𝑦)2 = 32

Write the standard form of the equation for the given information.

100. center (3,2) radius 6

101. center (-4, -7) radius 8

102. center (5, -9) radius 10

103. center (-8, 0) diameter 14

104. center (4,5) and point on the circle (3, -7)

105. diameter with endpoints (6, 4) and (10, -8)

106. center (4, 9) and tangent to the x-axis

PARCC-type Questions: Write the standard form of the equation for the given information.

107. 𝑥2 + 4𝑥 + 𝑦2 − 8𝑦 = 11

108. 𝑥2 − 10𝑥 + 𝑦2 + 2𝑦 = 11

109. 𝑥2 + 7𝑥 + 𝑦2 = 11

Are the following points on the circle (x-3)2+(y+4)2=25? Support your answer with your work.

110. (3,1)

111. (0,0)

112. (4,-1)

PARCC-type Question:

113. The equation 𝑥2 + 𝑦2 − 6𝑥 + 10𝑦 = 𝑏 describes a circle.

a. Determine the x-coordinate of the center of the circle.

b. Determine the y-coordinate of the center of the circle.

c. If the radius of the circle is 8 units, what is the value of b in the equation?

Equations of a Circle & Completing the Square: HOMEWORK

What are the center and the radius of the following circles?

114. (𝑥 − 9)2 + (𝑦 + 5)2 = 9

115. (𝑥 + 11)2 + (𝑦 − 8)2 = 64

116. (𝑥 + 13)2 + (𝑦 − 3)2 = 144

117. (𝑥 − 2)2 + (𝑦)2 = 19

118. (𝑥 − 6)2 + (𝑦 − 15)2 = 40

15

Write the standard form of the equation for the given information.

119. center (-2, -4) radius 9

120. center (-3, 3) radius 11

121. center (5, 8) radius 12

122. center (0 , 8) diameter 16

123. center (-4,6) and point on the circle (-2, -8)

124. diameter with endpoints (5, 14) and (11, -8)

125. center (4, 9) and tangent to the y-axis

PARCC-type Questions: Write the standard form of the equation for the given information.

126. 𝑥2 − 2𝑥 + 𝑦2 + 10𝑦 = 11

127. 𝑥2 + 12𝑥 + 𝑦2 + 20𝑦 = 11

128. 4𝑥2 + 16𝑥 + 4𝑦2 − 8𝑦 = 12

Are the following points on the circle (x-5)2+(y-12)2=169? Support your answer with your work.

129. (-4,2)

130. (0,0)

131. (-7,7)

PARCC-type Question:

132. The equation 𝑥2 + 𝑦2 + 12𝑥 − 4𝑦 = 𝑏 describes a circle.




16

Analytic Geometry Unit Review

Multiple Choice – Choose the correct answer for each question. No partial credit will be given.

1. What is the distance between points (-2, 1) and (2, 4)?

a. 3

b. c. 5

d.

2. What is the distance between points (1, 2) and (3, 4). Circle all that apply.

a. 2

b.

c. 2

d. 8

e. 2√5

f. 2√13

3. What is the midpoint between points (-1, 4) and (7, 6)

a. (6, 5)

b. (3, 5)

c. (5, 5)

d. (5, 3)

4. Find the midpoint between points (7, -9) and (3, 5)

a. (5, 2)

b. (2, 5)

c. (-5, 2)

d. (5, -2)

5. The midpoint of a line segment is (3, 4). One endpoint has the coordinates (-3, -2). What

are the coordinates of the other endpoint?

a. (9, 10)

b. (-3, -2)

c. (10, 9)

d. (1, 0)

6. If one line passes through the points (7, -3) & (-2, 3), what must be the slope of a line parallel

to it?

a. −3

2

b. −2

3

c. 0

d. undefined

17

7. If one line passes through the points (-5, 3) & (9, 7) and a perpendicular line passes through

the point (-1, -3), what is another point that would lie on the 2nd line? Circle all that apply.

a. (-3, 4)

b. (-3, -11)

c. (6, -1)

d. (6, -5)

e. (-10, 1)

f. (1, 13)

8. What is the equation of the line parallel to 2x + 8y = 10 and passes thru (-1, 5)?

a. y – 5 = - ¼(x – 1)

b. y – 5 = - ¼(x + 1)

c. y – 5 = 4(x – 1)

d. y – 5 = 4(x + 1)

9. What is the equation of the line perpendicular to y – 3 = 2/3(x + 3) and has an

x-intercept of 6?

a. y = 2/3x + 4

b. y = 2/3x – 4

c. y = -3/2x + 6

d. y = -3/2x + 9

10. What is the equation of the circle drawn in the figure

to the right?

a. (𝑥 − 6)2 + (𝑦 − 4)2 = 4

b. (𝑥 + 6)2 + (𝑦 + 4)2 = 6

c. (𝑥 − 6)2 + (𝑦 − 4)2 = 16

d. (𝑥 + 6)2 + (𝑦 + 4)2 = 36

Short Constructed Response – Write the correct answer for each question.

11. a) Write the distance formula and use it to find the distance between point B (-2, 5) to

point C (4, -3).

b) What are the coordinates of the midpoint of 𝐵𝐶̅̅ ̅̅ ?

12. The equation of a circle is 𝑥2 + 𝑦2 − 8𝑥 − 6𝑦 = 75. Write the equation in standard

form.

18

13. Line segment AB in the coordinate plane has endpoints with coordinates A (-11, -6) and

B (11, 5).


Extended Constructed Response – Solve the problem, showing all work.

14. Draw the line m on the graph provided so that m passes thru (1, 4) and (5, -3)

a. What is the equation of the line?

b. Construct a parallel line n that contains (3, 7).

c. What is the equation of line n?

d. Construct a line p that is perpendicular to the original line that contains A(3, 7).

e. What is the equation of line p?

b) Find 2 possible locations for

point C so that C divides 𝐴𝐵̅̅ ̅̅

into 2 parts with lengths in a

ratio of 6:5.

19

15. Given: The coordinates of

∆𝐺𝐻𝐽 & ∆𝐺𝐹𝑂

Prove: ∆𝐺𝐻𝐽 ≅ ∆𝐺𝐹𝑂

16. The points (3, 2) and (9, 12) are the endpoints of a diameter of a circle.

a. Where is the center of the circle?

b. How long is the diameter of the circle?

c. Write the equation of the circle?

d. Is the point (5, 6) inside, on, or outside the circle? Justify your answer.

17. The equation 𝑥2 + 𝑦2 − 8𝑥 + 6𝑦 = 𝑏 describes a circle.




Statements Reasons

1.

1.

2. 2. Vertical Angles

are

3. 𝑚𝐻𝐽 = ________

𝑚𝑂𝐹 = ________

3. Slope formula

4. 𝐻𝐽̅̅̅̅ || 𝑂𝐹̅̅ ̅̅ 4.

5. ∠𝐻 ≅∠𝐹 5.

6. ______ is the

midpoint of 𝐽𝑂̅̅ ̅

6. Midpoint formula

7. 𝑂𝐺̅̅ ̅̅ ≅ 𝐽𝐺̅̅ ̅ 7.

8. ∆𝐺𝐻𝐽 ≅ ∆𝐺𝐹𝑂 8.

GGGG

(3,

20

Answers: CW/HW problems:

1. 5

2. ≈ 10.82

3. √34 ≈ 5.83 4. 13

5. ≈ 4.24

6. ≈ 7.07 7. 13

8. ≈ 7.28

9. ≈ 5.10

10. ≈ 3.16 11. (3, 5)

12. (4, 5)

13. (1, 2)

14. (5, 1)

15. (-1.5, -10)

16. (4.5, 4)

17. (-2.5, 3)

18. (10, 16)

19. (-4, -4)

20. (1, 10)

21. (4, 2)

22. (3, 1)

23. (5, -2)

24. (4, 3.5)

25. (-4, -4)

26. (3.5, 1)

27. (-1.5, 0.5)

28. (-11, -13)

29. (14, -21)

30. (-4, 10)

31. See graph below for a) & b)


33. (-1, -1) & (2, 2)

34. (2, -4) & (-8, 6)

35. (-3, 4) & (1, -2)

3 13

34

3 2

5 2

53

26

10

21



38. (-4, -4) & (5, 5)

39. (-4, 2) & (-2, 0)

40. (-7, 10) & (5, -8)

41. 3

42. No slope (undefined)

43. 0

44. a. no; slope of j=2, slope of k = 4/3;

b. yes; slope of j and k = -4/5

45. a. yes; slope of j= -1/6; slope of k=6;

b. no slope of j= -1; slope of k= 1/3

46. m = -5

47. ½

48. Multiple Answers: Sample points

that work include (4, -3) & (2, 11)

49. -2

50. 1/7


that work include (1, 5) & (5, 3)

52. -7/4

53. 0

54. No slope (Undefined)

55. a. yes; slope of j & k = 4/7;

b. no; slope of j= -1/6, slope of k= -1/7

56. a. no slope of j= 3/2; slope of k=-1/2;

b. yes, slope of j=1/5; slope of k= -5

57. ½

58. -7

59. Multiple answers: Sample points that

work include (2, 2) & (0, -2)

60. m = 1/5

61. -2


that work include (-4, 3) or (10, 5)

63. y -5= 2(x+2)

64. k= -1/2

65. y= -3/4x+3

66. Yes,𝑦 = 4𝑥 + 5䚫8𝑥 − 2𝑦 =

1 ; 𝑦 =1

3𝑥 + 7䚫𝑥 − 3𝑦 = 2

67. y=-2x+3

68. k= -2

69. y=-1/4x

70. no, y=4x +5 and y =1/3x +7 are not

perpendicular

71. perpendicular

72. neither

73. y -2 = 3(x+3)

74. k=1/3

75. y-3=-3/4(x+8)

76. no;

77. y=3/2x+11

78. k=-3/2

79. y=1/2x-6

80. yes, slopes are m=-1/4 and 4

81. parallel

82. neither

83. a.) 2√5 or 4.5 b.) (5, 5)

84. a.) √73 or 8.5 b.) (0.5, 1)

85. a.) -3 b.) -1/3

22

86. a.) ½ b.) -2 87. Statements Reasons

1. 𝐺𝐽⃗⃗⃗⃗ bisects OGH 1. Given

2. OGJ HGJ 2. Definition of an

angle bisector

3. 𝐺𝐽̅̅ ̅ ≅ 𝐺𝐽̅̅ ̅ 3. Reflexive Prop. of

4. 𝑂𝐺 = √34,

𝐻𝐺 = √34

4. Distance Formula

5. 𝑂𝐺̅̅ ̅̅ ≅ 𝐻𝐺̅̅ ̅̅ 5. Def. of

segments

6. GJO GJH 6. SAS Postulate

88.

Statements Reasons

1. Coordinates of

vertices of OPM

and ONM

1. Given

2. OP = 5, PM = 5,

MN = 5, NO = 5

2. Distance Formula

3. 𝑂𝑃̅̅ ̅̅ ≅ 𝑃𝑀̅̅̅̅̅ ≅𝑀𝑁̅̅ ̅̅ ̅ ≅ 𝑁𝑂̅̅ ̅̅

3. Def. of

segments

4. 𝑂𝑀̅̅ ̅̅ ̅ ≅ 𝑂𝑀̅̅ ̅̅ ̅ 4. Reflexive Prop. of

5. OPM and

ONM are isosceles

5. Definition of an

isosceles triangle

6. OPM ONM 6. SSS Postulate

89. a.) √85 or 9.2 b.) (10.5, -5) 90. a.) 13 b.) (-2.5, 6) 91. a.) 2/7 b.) -7/2 92. a.) -2/3 b.) 3/2 93. Statements Reasons

1. 𝑂𝑆̅̅̅̅ ⊥ 𝑅𝑇̅̅ ̅̅ 1. Given

2. RSO and TSO

are right angles

2. Definition of ⊥

lines

3. RSO ≅ TSO 3. All right angles

are congruent

4. 𝑂𝑆̅̅̅̅ ≅ 𝑂𝑆̅̅̅̅ 4. Reflexive Prop. of

5. OR = 6, OT = 6 5. Given in diagram;

or Distance formua

6. 𝑂𝑅̅̅ ̅̅ ≅ 𝑂𝑇̅̅ ̅̅ 6. Def. of

segments

7. SOR SOT 7. HL Theorem

8. SOR SOT 8. CPCTC

9. 𝑂𝑆⃗⃗⃗⃗ ⃗ bisects TOR 9. Definition of an

angle bisector

94. Statements Reasons

1. G is the midpoint

of 𝐻𝐹̅̅ ̅̅

1. Given

2. HGJ FGO 2. Vertical Angles

are

3. 𝐻𝐺̅̅ ̅̅ ≅ 𝐹𝐺̅̅ ̅̅ 3. Def. of midpoint

4. 𝑂𝐺 = 3√2

𝐽𝐺 = 3√2

4. Distance Formula

5. 𝑂𝐺̅̅ ̅̅ ≅ 𝐽𝐺̅̅ ̅ 5. Definition of

segments

6. GHJ GFO 6. SAS Postulate

95. C(-2,4); r=4

96. C (3,7); r=5

97. C (),-8); r=1

98. C (7,-1); r= √17

99. C (-6,0); r =4√2

100. (x-3)2 + (x-2)2 =36

101. (x+4)2 + (Y+7)2 =64

102. (x-5)2 + (y+9)2 = 100

103. (x+8)2 + y2 =49

104. (x-4)2 + (y-5)2 =145

105. (x-8)2 + (y+2)2 =40

106. (x-4)2 + (y-9)2 =81

107. (x+2)2 + (y-4)2 =31

108. (x-5)2 + (y+1)2 =37

109. (x+3.5)2 + y2 =23.25

110. yes; (3-3)2+(1+4)2=25

111. yes; (0-3)2+(0+4)2=25

112. no; (4-3)2+(-1+4)2=10

113. a) x-coord. = 3

b) y-coord. = -5

c) b = 30

114. C (9,-5) r=3

23

115. C -11, 8) r=8

116. C(-13, 3) r=12

117. C(2,0) r= √19

118. C (6,15) r=2√10

119. (x+2)2 +(Y+4)2 =81

120. (x+3)2 + (y-3)2 =121

121. (x-5)2 + (y-8)2 =144

122. x2 + (y-8)2 =64

123. (x+4)2 + (y-6)2 =200

124. (x-8)2 + (y-3)2 = 130

125. (x-4)2 + ( y-9)2 =130

126. (x-1)2 + (y+5)2 =37

127. (x+6)2 + (y+10)2 =147

128. (x+2)2 + (y-1)2 =8

129. no; (-4-5)2+(2-12)2=181

130. yes; (0-5)2+(0-12)2=169

131. yes; (-7-5)2+(7-12)2=169

132. a) x-coord. = -6

b) y-coord. = 2

c) b = -15

Answers: Unit Review

Multiple Choice

1. C

2. B & C

3. B

4. D

5. A

6. B

7. A & E

8. B

9. D

10. C

Short Constructed Response

11. d = 10; midpoint = (1, 1)

12. (𝑥 − 4)2 + (𝑦 − 3)2 = 100 13. See graph below for a) & b)

Extended Constructed Response

14. a. y – 4 = -7/4(x – 1)

or y + 3 = -7/4(x – 5)

b. construct 𝑦 − 7 = −7/4(𝑥 − 3)

in coordinate plane

c. y – 7 = -7/4(x – 3) or

y = -7/4x + 49/4 or

7x + 4y = 49

d. construct 𝑦 − 7 = 4/7(𝑥 − 3)

in coordinate plane

e. 𝑦 − 7 = 4/7(𝑥 − 3) or

y = 4/7x + 37/7 or

4x – 7y = -37

15.

Statements Reasons

1. The coordinates of

∆𝐺𝐻𝐽 & ∆𝐺𝐹𝑂

1. Given

2. HGJ FGO 2. Vertical Angles

are

3. 𝑚𝐻𝐽 = 0

𝑚𝑂𝐹 = 0

3. Slope formula, or

Horizontal lines

have a slope of 0.

4. 𝐻𝐽̅̅̅̅ || 𝑂𝐹̅̅ ̅̅ 4. Def. of || lines

5. ∠𝐻 ≅∠𝐹 5. Alt. Int. ∠𝑠 ≅

6. G is the midpoint

of 𝐽𝑂̅̅ ̅

6. Midpoint formula

7. 𝑂𝐺̅̅ ̅̅ ≅ 𝐽𝐺̅̅ ̅ 7. Definition of

midpoint

8. GHJ GFO 8. AAS Theorem

24

16. a. (6,7)

b. 11.662

c. (𝑥 − 6)2+(𝑦 − 7)2 = 136

d. (5 − 6)2+(6 − 7)2 = 2

2 < 136, so the point is inside the

circle.

17. Equation:

(𝑥 − 4)2 + (𝑦 + 3)2 = 𝑏 + 25 a. 4

b. -3

c. b = 75, since 𝑟2 = 100

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Analytic Geometry - content.njctl.orgcontent.njctl.org › courses › math › geometry-2015-16 › ... · Geometry – Analytic Geometry ~2~ NJCTL.org Midpoint Formula: CLASSWORK