Name: ______________________ Date: ____________ Ms. Ayinde Geometry CC 5.1: Dilations

Objective: To draw dilations in the coordinate plane. To determine the equation of a line after being dilated. Our next unit focuses on similarity. Similar figures are figures that have congruent corresponding angles, but the corresponding sides are proportional. Basically, these figures are the same shape, but different size. Unlike in previous units, we need to focus on ratios and proportions. Let’s take a closer look at similarities in the coordinate plane. A dilation is a transformation that produces an image that is the same shape as the original but not the same size. A dilation is not a rigid motion.

● A dilation is similar to the original figure. ○ Similare figures are polygons that have congruent corresponding angles and proportional sides.

● Dilations are centered around the origin (0, 0), unless otherwise stated. Scale factor – is image length , which is a scale factor. pre- image length

● If the scale factor is greater than 1, the figure becomes enlarged.

● If the scale factor is between 0 and 1, the figure becomes reduced. Rule: Multiply the coordinate by the scale factor. Example 1: Triangle ABC has vertices A (0, 2), B (4, 4), and C (-1, 4). What are the vertices of its image with a scale factor of 4?

Example 2: A line segment dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?

(1) The line segments are perpendicular, and the image is one-half of the length of the given line segment.

(2) The line segments are perpendicular, and the image is twice the length of the given line segment.

(3) The line segments are parallel, the image is twice the length of the given line.

(4) The line segments are parallel, the image is one-half of the length of the given line segment.

Example 3: Quadrilateral PQRS has vertices P (-2, 4), Q (4, 4), R (4, -2), and S (- 4, - 4). It is dilated by a scale factor of ½. a) What are the coordinates of the image? Graph them.

b) Demonstrate these quadrilaterals are similar by comparing the ratios of the lengths. c) What do you notice about the angle measurements of the two figures? Example 4: Quadrilateral A’B’C’D’ is a dilation of quadrilateral ABCD. Find the scale factor. Classify the dilation as an enlargement or a reduction.

Practice Problems: 1.

2.

3.

4.

You can also dilate lines in the coordinate plane. Example 5

Example 6

Example 7

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Geometry Regents Exam Questions by Common Core State Standard: Topic

215 The equation of line h is 2x � y 1. Line m is the image of line h after a dilation of scale factor 4 with respect to the origin. What is the equation of the line m?1) y �2x � 12) y �2x � 43) y 2x � 44) y 2x � 1

216 The line y 2x � 4 is dilated by a scale factor of 32

and centered at the origin. Which equation represents the image of the line after the dilation?1) y 2x � 42) y 2x � 63) y 3x � 44) y 3x � 6

217 The line 3y �2x � 8 is transformed by a dilation centered at the origin. Which linear equation could be its image?1) 2x � 3y 52) 2x � 3y 53) 3x � 2y 54) 3x � 2y 5

218 Line y 3x � 1 is transformed by a dilation with a scale factor of 2 and centered at (3,8). The line's image is1) y 3x � 82) y 3x � 43) y 3x � 24) y 3x � 1

219 On the graph below, point A(3,4) and BC with coordinates B(4,3) and C(2,1) are graphed.

What are the coordinates of B’ and C’ after BC undergoes a dilation centered at point A with a scale factor of 2?1) B' (5,2) and C ' (1,�2)2) B' (6,1) and C ' (0,�1)3) B' (5,0) and C ' (1,�2)4) B' (5,2) and C ' (3,0)

220 A line that passes through the points whose coordinates are (1,1) and (5,7) is dilated by a scale factor of 3 and centered at the origin. The image of the line1) is perpendicular to the original line2) is parallel to the original line3) passes through the origin4) is the original line

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

51

221 A three-inch line segment is dilated by a scale factor of 6 and centered at its midpoint. What is the length of its image?1) 9 inches2) 2 inches3) 15 inches4) 18 inches

222 Line segment A'B', whose endpoints are (4,�2) and

(16,14), is the image of AB after a dilation of 12

centered at the origin. What is the length of AB?1) 52) 103) 204) 40

223 A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?1) The line segments are perpendicular, and the

image is one-half of the length of the given line segment.

2) The line segments are perpendicular, and the image is twice the length of the given line segment.

3) The line segments are parallel, and the image is twice the length of the given line segment.

4) The line segments are parallel, and the image is one-half of the length of the given line segment.

224 The line represented by the equation 4y 3x � 7 is transformed by a dilation centered at the origin. Which linear equation could represent its image?1) 3x � 4y 92) 3x � 4y 93) 4x � 3y 94) 4x � 3y 9

225 Line n is represented by the equation 3x � 4y 20. Determine and state the equation of line p, the

image of line n, after a dilation of scale factor 13

centered at the point (4,2). [The use of the set of axes below is optional.] Explain your answer.

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

50

Geometry Regents Exam Questions by Common Core State Standard: Topic

215 The equation of line h is 2x � y 1. Line m is the image of line h after a dilation of scale factor 4 with respect to the origin. What is the equation of the line m?1) y �2x � 12) y �2x � 43) y 2x � 44) y 2x � 1

216 The line y 2x � 4 is dilated by a scale factor of 32

and centered at the origin. Which equation represents the image of the line after the dilation?1) y 2x � 42) y 2x � 63) y 3x � 44) y 3x � 6

217 The line 3y �2x � 8 is transformed by a dilation centered at the origin. Which linear equation could be its image?1) 2x � 3y 52) 2x � 3y 53) 3x � 2y 54) 3x � 2y 5

218 Line y 3x � 1 is transformed by a dilation with a scale factor of 2 and centered at (3,8). The line's image is1) y 3x � 82) y 3x � 43) y 3x � 24) y 3x � 1

219 On the graph below, point A(3,4) and BC with coordinates B(4,3) and C(2,1) are graphed.

What are the coordinates of B’ and C’ after BC undergoes a dilation centered at point A with a scale factor of 2?1) B' (5,2) and C ' (1,�2)2) B' (6,1) and C ' (0,�1)3) B' (5,0) and C ' (1,�2)4) B' (5,2) and C ' (3,0)

220 A line that passes through the points whose coordinates are (1,1) and (5,7) is dilated by a scale factor of 3 and centered at the origin. The image of the line1) is perpendicular to the original line2) is parallel to the original line3) passes through the origin4) is the original line

Practice Problems: 5.

6.

7.

8.

9.

10.

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

50

Geometry Regents Exam Questions by Common Core State Standard: Topic

215 The equation of line h is 2x � y 1. Line m is the image of line h after a dilation of scale factor 4 with respect to the origin. What is the equation of the line m?1) y �2x � 12) y �2x � 43) y 2x � 44) y 2x � 1

216 The line y 2x � 4 is dilated by a scale factor of 32

and centered at the origin. Which equation represents the image of the line after the dilation?1) y 2x � 42) y 2x � 63) y 3x � 44) y 3x � 6

217 The line 3y �2x � 8 is transformed by a dilation centered at the origin. Which linear equation could be its image?1) 2x � 3y 52) 2x � 3y 53) 3x � 2y 54) 3x � 2y 5

218 Line y 3x � 1 is transformed by a dilation with a scale factor of 2 and centered at (3,8). The line's image is1) y 3x � 82) y 3x � 43) y 3x � 24) y 3x � 1

219 On the graph below, point A(3,4) and BC with coordinates B(4,3) and C(2,1) are graphed.

What are the coordinates of B’ and C’ after BC undergoes a dilation centered at point A with a scale factor of 2?1) B' (5,2) and C ' (1,�2)2) B' (6,1) and C ' (0,�1)3) B' (5,0) and C ' (1,�2)4) B' (5,2) and C ' (3,0)

220 A line that passes through the points whose coordinates are (1,1) and (5,7) is dilated by a scale factor of 3 and centered at the origin. The image of the line1) is perpendicular to the original line2) is parallel to the original line3) passes through the origin4) is the original line

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

50

Geometry Regents Exam Questions by Common Core State Standard: Topic

215 The equation of line h is 2x � y 1. Line m is the image of line h after a dilation of scale factor 4 with respect to the origin. What is the equation of the line m?1) y �2x � 12) y �2x � 43) y 2x � 44) y 2x � 1

216 The line y 2x � 4 is dilated by a scale factor of 32

and centered at the origin. Which equation represents the image of the line after the dilation?1) y 2x � 42) y 2x � 63) y 3x � 44) y 3x � 6

217 The line 3y �2x � 8 is transformed by a dilation centered at the origin. Which linear equation could be its image?1) 2x � 3y 52) 2x � 3y 53) 3x � 2y 54) 3x � 2y 5

218 Line y 3x � 1 is transformed by a dilation with a scale factor of 2 and centered at (3,8). The line's image is1) y 3x � 82) y 3x � 43) y 3x � 24) y 3x � 1

219 On the graph below, point A(3,4) and BC with coordinates B(4,3) and C(2,1) are graphed.

What are the coordinates of B’ and C’ after BC undergoes a dilation centered at point A with a scale factor of 2?1) B' (5,2) and C ' (1,�2)2) B' (6,1) and C ' (0,�1)3) B' (5,0) and C ' (1,�2)4) B' (5,2) and C ' (3,0)

220 A line that passes through the points whose coordinates are (1,1) and (5,7) is dilated by a scale factor of 3 and centered at the origin. The image of the line1) is perpendicular to the original line2) is parallel to the original line3) passes through the origin4) is the original line

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

50

Geometry Regents Exam Questions by Common Core State Standard: Topic

215 The equation of line h is 2x � y 1. Line m is the image of line h after a dilation of scale factor 4 with respect to the origin. What is the equation of the line m?1) y �2x � 12) y �2x � 43) y 2x � 44) y 2x � 1

216 The line y 2x � 4 is dilated by a scale factor of 32

and centered at the origin. Which equation represents the image of the line after the dilation?1) y 2x � 42) y 2x � 63) y 3x � 44) y 3x � 6

217 The line 3y �2x � 8 is transformed by a dilation centered at the origin. Which linear equation could be its image?1) 2x � 3y 52) 2x � 3y 53) 3x � 2y 54) 3x � 2y 5

218 Line y 3x � 1 is transformed by a dilation with a scale factor of 2 and centered at (3,8). The line's image is1) y 3x � 82) y 3x � 43) y 3x � 24) y 3x � 1

219 On the graph below, point A(3,4) and BC with coordinates B(4,3) and C(2,1) are graphed.

What are the coordinates of B’ and C’ after BC undergoes a dilation centered at point A with a scale factor of 2?1) B' (5,2) and C ' (1,�2)2) B' (6,1) and C ' (0,�1)3) B' (5,0) and C ' (1,�2)4) B' (5,2) and C ' (3,0)

220 A line that passes through the points whose coordinates are (1,1) and (5,7) is dilated by a scale factor of 3 and centered at the origin. The image of the line1) is perpendicular to the original line2) is parallel to the original line3) passes through the origin4) is the original line

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

51

221 A three-inch line segment is dilated by a scale factor of 6 and centered at its midpoint. What is the length of its image?1) 9 inches2) 2 inches3) 15 inches4) 18 inches

222 Line segment A'B', whose endpoints are (4,�2) and

(16,14), is the image of AB after a dilation of 12

centered at the origin. What is the length of AB?1) 52) 103) 204) 40

223 A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?1) The line segments are perpendicular, and the

image is one-half of the length of the given line segment.

2) The line segments are perpendicular, and the image is twice the length of the given line segment.

3) The line segments are parallel, and the image is twice the length of the given line segment.

4) The line segments are parallel, and the image is one-half of the length of the given line segment.

224 The line represented by the equation 4y 3x � 7 is transformed by a dilation centered at the origin. Which linear equation could represent its image?1) 3x � 4y 92) 3x � 4y 93) 4x � 3y 94) 4x � 3y 9

225 Line n is represented by the equation 3x � 4y 20. Determine and state the equation of line p, the

image of line n, after a dilation of scale factor 13

centered at the point (4,2). [The use of the set of axes below is optional.] Explain your answer.

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

51

221 A three-inch line segment is dilated by a scale factor of 6 and centered at its midpoint. What is the length of its image?1) 9 inches2) 2 inches3) 15 inches4) 18 inches

222 Line segment A'B', whose endpoints are (4,�2) and

(16,14), is the image of AB after a dilation of 12

centered at the origin. What is the length of AB?1) 52) 103) 204) 40

223 A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?1) The line segments are perpendicular, and the

image is one-half of the length of the given line segment.

2) The line segments are perpendicular, and the image is twice the length of the given line segment.

3) The line segments are parallel, and the image is twice the length of the given line segment.

4) The line segments are parallel, and the image is one-half of the length of the given line segment.

224 The line represented by the equation 4y 3x � 7 is transformed by a dilation centered at the origin. Which linear equation could represent its image?1) 3x � 4y 92) 3x � 4y 93) 4x � 3y 94) 4x � 3y 9

225 Line n is represented by the equation 3x � 4y 20. Determine and state the equation of line p, the

image of line n, after a dilation of scale factor 13

centered at the point (4,2). [The use of the set of axes below is optional.] Explain your answer.

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

51

221 A three-inch line segment is dilated by a scale factor of 6 and centered at its midpoint. What is the length of its image?1) 9 inches2) 2 inches3) 15 inches4) 18 inches

222 Line segment A'B', whose endpoints are (4,�2) and

(16,14), is the image of AB after a dilation of 12

centered at the origin. What is the length of AB?1) 52) 103) 204) 40

223 A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?1) The line segments are perpendicular, and the

image is one-half of the length of the given line segment.

2) The line segments are perpendicular, and the image is twice the length of the given line segment.

3) The line segments are parallel, and the image is twice the length of the given line segment.

4) The line segments are parallel, and the image is one-half of the length of the given line segment.

224 The line represented by the equation 4y 3x � 7 is transformed by a dilation centered at the origin. Which linear equation could represent its image?1) 3x � 4y 92) 3x � 4y 93) 4x � 3y 94) 4x � 3y 9

225 Line n is represented by the equation 3x � 4y 20. Determine and state the equation of line p, the

image of line n, after a dilation of scale factor 13

centered at the point (4,2). [The use of the set of axes below is optional.] Explain your answer.

Name: ______________________ Date: ____________ Ms. Ayinde Geometry CC 5.2: Partitioning of a Line Segment Objective: To determine the coordinates of each point of partition. Do Now: What partition is Beyoncé referring to in her song, Partition?

Example 1

Using Similar Triangles

Using a Formula

Practice:

1.

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

8

26 The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

LINES AND ANGLESG.GPE.B.6: DIRECTED LINE SEGMENTS

27 What are the coordinates of the point on the directed line segment from K(−5,−4) to L(5,1) that partitions the segment into a ratio of 3 to 2?1) (−3,−3)2) (−1,−2)

3) 0,−32

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4) (1,−1)

28 Point P is on segment AB such that AP :PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.

29 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE :EF = 2:3.

30 The coordinates of the endpoints of AB are A(−6,−5) and B(4,0). Point P is on AB. Determine and state the coordinates of point P, such that AP :PB is 2:3. [The use of the set of axes below is optional.]

31 Point Q is on MN such that MQ:QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,−5), the coordinates of Q are1) (5,1)2) (5,0)3) (6,−1)4) (6,0)

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

8

26 The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

LINES AND ANGLESG.GPE.B.6: DIRECTED LINE SEGMENTS

27 What are the coordinates of the point on the directed line segment from K(−5,−4) to L(5,1) that partitions the segment into a ratio of 3 to 2?1) (−3,−3)2) (−1,−2)

3) 0,−32

ä

æ

åååååå

ô

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õõõõõõ

4) (1,−1)

28 Point P is on segment AB such that AP :PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.

29 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE :EF = 2:3.

30 The coordinates of the endpoints of AB are A(−6,−5) and B(4,0). Point P is on AB. Determine and state the coordinates of point P, such that AP :PB is 2:3. [The use of the set of axes below is optional.]

31 Point Q is on MN such that MQ:QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,−5), the coordinates of Q are1) (5,1)2) (5,0)3) (6,−1)4) (6,0)

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

8

26 The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

LINES AND ANGLESG.GPE.B.6: DIRECTED LINE SEGMENTS

27 What are the coordinates of the point on the directed line segment from K(−5,−4) to L(5,1) that partitions the segment into a ratio of 3 to 2?1) (−3,−3)2) (−1,−2)

3) 0,−32

ä

æ

åååååå

ô

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õõõõõõ

4) (1,−1)

28 Point P is on segment AB such that AP :PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.

29 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE :EF = 2:3.

30 The coordinates of the endpoints of AB are A(−6,−5) and B(4,0). Point P is on AB. Determine and state the coordinates of point P, such that AP :PB is 2:3. [The use of the set of axes below is optional.]

31 Point Q is on MN such that MQ:QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,−5), the coordinates of Q are1) (5,1)2) (5,0)3) (6,−1)4) (6,0)

2.

3.

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

8

26 The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

LINES AND ANGLESG.GPE.B.6: DIRECTED LINE SEGMENTS

27 What are the coordinates of the point on the directed line segment from K(−5,−4) to L(5,1) that partitions the segment into a ratio of 3 to 2?1) (−3,−3)2) (−1,−2)

3) 0,−32

ä

æ

åååååå

ô

ö

õõõõõõ

4) (1,−1)

28 Point P is on segment AB such that AP :PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.

29 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE :EF = 2:3.

30 The coordinates of the endpoints of AB are A(−6,−5) and B(4,0). Point P is on AB. Determine and state the coordinates of point P, such that AP :PB is 2:3. [The use of the set of axes below is optional.]

31 Point Q is on MN such that MQ:QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,−5), the coordinates of Q are1) (5,1)2) (5,0)3) (6,−1)4) (6,0)

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

8

26 The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

LINES AND ANGLESG.GPE.B.6: DIRECTED LINE SEGMENTS

27 What are the coordinates of the point on the directed line segment from K(−5,−4) to L(5,1) that partitions the segment into a ratio of 3 to 2?1) (−3,−3)2) (−1,−2)

3) 0,−32

ä

æ

åååååå

ô

ö

õõõõõõ

4) (1,−1)

28 Point P is on segment AB such that AP :PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.

29 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE :EF = 2:3.

30 The coordinates of the endpoints of AB are A(−6,−5) and B(4,0). Point P is on AB. Determine and state the coordinates of point P, such that AP :PB is 2:3. [The use of the set of axes below is optional.]

31 Point Q is on MN such that MQ:QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,−5), the coordinates of Q are1) (5,1)2) (5,0)3) (6,−1)4) (6,0)

4.

5.

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

8

26 The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

LINES AND ANGLESG.GPE.B.6: DIRECTED LINE SEGMENTS

27 What are the coordinates of the point on the directed line segment from K(−5,−4) to L(5,1) that partitions the segment into a ratio of 3 to 2?1) (−3,−3)2) (−1,−2)

3) 0,−32

ä

æ

åååååå

ô

ö

õõõõõõ

4) (1,−1)

28 Point P is on segment AB such that AP :PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.

29 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE :EF = 2:3.

30 The coordinates of the endpoints of AB are A(−6,−5) and B(4,0). Point P is on AB. Determine and state the coordinates of point P, such that AP :PB is 2:3. [The use of the set of axes below is optional.]

31 Point Q is on MN such that MQ:QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,−5), the coordinates of Q are1) (5,1)2) (5,0)3) (6,−1)4) (6,0)

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

8

26 The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

LINES AND ANGLESG.GPE.B.6: DIRECTED LINE SEGMENTS

27 What are the coordinates of the point on the directed line segment from K(−5,−4) to L(5,1) that partitions the segment into a ratio of 3 to 2?1) (−3,−3)2) (−1,−2)

3) 0,−32

ä

æ

åååååå

ô

ö

õõõõõõ

4) (1,−1)

28 Point P is on segment AB such that AP :PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.

29 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE :EF = 2:3.

30 The coordinates of the endpoints of AB are A(−6,−5) and B(4,0). Point P is on AB. Determine and state the coordinates of point P, such that AP :PB is 2:3. [The use of the set of axes below is optional.]

31 Point Q is on MN such that MQ:QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,−5), the coordinates of Q are1) (5,1)2) (5,0)3) (6,−1)4) (6,0)Geometry Regents Exam Questions by State Standard: Topic

www.jmap.org

9

32 Point P is on the directed line segment from point X(−6,−2) to point Y(6,7) and divides the segment in the ratio 1:5. What are the coordinates of point P?

1) 4,5 12

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2) −12 ,−4

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3) −4 12 ,0

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4) −4,−12

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33 Directed line segment PT has endpoints whose coordinates are P(−2,1) and T(4,7). Determine the coordinates of point J that divides the segment in the ratio 2 to 1. [The use of the set of axes below is optional.]

34 In the diagram below, AC has endpoints with coordinates A(−5,2) and C(4,−10).

If B is a point on AC and AB :BC = 1:2, what are the coordinates of B?1) (−2,−2)

2) −12 ,−4

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õõõõõõ

3) 0,−143

ä

æ

åååååå

ô

ö

õõõõõõ

4) (1,−6)

35 Line segment RW has endpoints R(−4,5) and W(6,20). Point P is on RW such that RP:PW is 2:3. What are the coordinates of point P?1) (2,9)2) (0,11)3) (2,14)4) (10,2)

Geometry Regents Exam Questions by State Standard: Topicwww.jmap.org

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26 The diagram below shows circle O with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle O. [Leave all construction marks.]

LINES AND ANGLESG.GPE.B.6: DIRECTED LINE SEGMENTS

27 What are the coordinates of the point on the directed line segment from K(−5,−4) to L(5,1) that partitions the segment into a ratio of 3 to 2?1) (−3,−3)2) (−1,−2)

3) 0,−32

ä

æ

åååååå

ô

ö

õõõõõõ

4) (1,−1)

28 Point P is on segment AB such that AP :PB is 4:5. If A has coordinates (4,2), and B has coordinates (22,2), determine and state the coordinates of P.

29 The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE :EF = 2:3.

30 The coordinates of the endpoints of AB are A(−6,−5) and B(4,0). Point P is on AB. Determine and state the coordinates of point P, such that AP :PB is 2:3. [The use of the set of axes below is optional.]

31 Point Q is on MN such that MQ:QN = 2:3. If M has coordinates (3,5) and N has coordinates (8,−5), the coordinates of Q are1) (5,1)2) (5,0)3) (6,−1)4) (6,0)