Post on 06-Jun-2020
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).
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 1:4.
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
72. Determine if the following equations are parallel, perpendicular, or neither. Justify your
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.
81. Determine if the following equations are parallel, perpendicular, or neither. Justify your
answer. 9x + 5y = 32 4.5x + 2.5y = 7.5
82. Determine if the following equations are parallel, perpendicular, or neither. Justify your
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
For numbers 89 – 90 find (a.) the length of and (b.) the midpoint coordinates of .
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
line perpendicular to it?
92. If one line passes through the points (-2,5) and (7,-1) what must be the slope of (a.) any line
parallel to that first line and (b.) any line perpendicular to that first 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
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
13
Triangle Coordinate Proofs: HOMEWORK – Honors
For numbers 89 – 90 find (a.) the length of and (b.) the midpoint coordinates of .
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
line perpendicular to it?
92. If one line passes through the points (-2,5) and (7,-1) what must be the slope of (a.) any line
parallel to that first line and (b.) any line perpendicular to that first 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.
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 5 units, what is the value of b in the equation?
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).
a) Graph 𝐴𝐵̅̅ ̅̅
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.
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 10 units, what is the value of b in the equation?
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)
32. 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
36. See graph below for a) & b)
37. See graph below for a) & b)
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
51. Multiple Answers: Sample points
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
62. Multiple Answers: Sample points
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