Teacher Page

Geometry / Day # 10 Coordinate Geometry (45 min.) 9-12.G.3.1 Use rigid motions (compositions of reflections, translations and rotations) to

determine whether two geometric figures are congruent in a coordinate plane. 9-12.G.3.2 Sketch a planar figure that is the result of given transformations (i.e. translation,

reflection, rotation, and/or dilation) 9-12.G.3.3 Identify similarity in terms of transformations

9-12.G.3.4 Determine the effects of transformations on linear and area measurements of the original planar figure.

1. Reflect a figure with vertices across the x-axis. Find the coordinates of the new image. a. b. c. d. 2. vertices A(3, 1), B(4, 5), and C(2, 3). Rotate 90° about the origin and then reflect it across the x-axis. a.

A

B

CA'B'

C'

A"

B"

C"x

y c.

A

B

C

A'

B'

C'

A"

B"

C"

x

y

b.

A

B

C

A'B'

C'

A"

B"

C"x

y d.

A

B

C

A'

B'

C'A"

B"

C"x

y

3. A triangle is defined by the points (3, 4), (10, -5), and (-9, -1). Sketch the triangle. Find the coordinates of the points after the triangle has been translated 7 units up.

5 10–5–10 x

5

10

–5

–10

y

In the answer box provided, with words, graphs, tables or equations, show your solution to the

problem. Only work within the answer box will be scored.

3.

4. Graph the figure with coordinates at A(8,10), B(6,-6), C(-3,-4), and D(-5,6).

A. Sketch its image under a dilation of 12

centered at the origin.

B. Find the coordinates

5 10–5–10 x

5

10

–5

–10

y

In the answer box provided, with words, graphs, tables or equations, show your solution to the

problem. Only work within the answer box will be scored. 4.

Key Teacher Page Geometry / Day #10 Coordinate Geometry (45 min.)

1. Reflect a figure with vertices across the x-axis. Find the coordinates of the new image. a. b. (Correct Answer) c. d. Solution: The reflection of across the x-axis is .

Feedback A The reflection of (x, y) on the x-axis is (x, –y). B Correct! C The reflection of (x, y) on the x-axis is (x, –y). D The reflection of (x, y) on the x-axis is (x, –y).

2. vertices A(3, 1), B(4, 5), and C(2, 3). Rotate 90° about the origin and then reflect it across the x-axis. (Correct Answer) a.

A

B

CA'B'

C'

A"

B"

C"x

y c.

A

B

C

A'

B'

C'

A"

B"

C"

x

y

b.

A

B

C

A'B'

C'

A"

B"

C"x

y d.

A

B

C

A'

B'

C'A"

B"

C"x

y

The rotation image of (x, y) 90° counterclockwise is (–y, x), so A(3, 1) (–1, 3), B(4, 5) (–5, 4) and C(2, 3) (–3, 2). The reflection image of (x, y) across the x-axis is (x, –y), so

(–1, 3) (–1, –3), (–5, 4) (–5, –4) and (–3, 2) (–3, –2). Feedback A Correct! B This is a rotation of 90° clockwise about the origin and a reflection is across the y-axis. C This is a rotation of 180° about the origin, and then a reflection across the x-axis. D This is a reflection across the x-axis, and then a rotation of 90° clockwise about the origin.

3. A triangle is defined by the points (3, 4), (10, -5), and (-9, -1). Sketch the triangle. Find the coordinates of the points after the triangle has been translated 7 units up.

In the answer box provided, with words, graphs, tables or equations, show your solution to the

problem. Only work within the answer box will be scored.

3.

(3, 4)

(10, –5)

(–9, –1) 5 10–5–10 x

5

10

–5

–10

y

Solution: Original Triangle Sketched (1 Point) Translated triangle vertices (1 Point)

(3, 11) (10, 2) (-9, 6)

4. Graph the figure with coordinates at A(8,10), B(6,-6), C(-3,-4), and D(-5,6). In the answer box provided, with words, graphs, tables or equations, show your solution to the

problem. Only work within the answer box will be scored.

4.

Solution: Sketch Original Image (1 Point)

A. Sketch its image under a dilation of 12

centered at the origin. (1 Point)

B. Find the coordinates (2 Points) These points are obtained by multiplying all original coordinates by ½. .

A’(4, 5) B’(3,-3) C’(-1.5,-2) D’(-2.5, 3)

Name:_____________________________ Date:______________________ Per:_____ Student Page: Geometry/ Day #10 Coordinate Geometry

1. Ann wants to create a design to decorate her Geometry binder. She reflects part of the design across line p and then reflects the image across line n. Describe a single transformation that moves the part of the design from its starting position to its final position.

Start

Finish

n

p

a. rotation of 180° about the origin c. translation along the line b. rotation of 90° about the origin d. reflection across the line

2. Given the rectangle and the center of dilation P, which of the following is dilation with a scale factor of 2?

P

a.

P

c.

P

b.

P

d.

P

3. In a dance performance, four dancers form a diamond with vertices

. Then, they move along the dance floor following the translation

vector, . There they pause, and then move again along the same vector. What are their coordinates after six such translations? a. b. c. d.

4. Describe the dilation in the graph.

A B

C

A' B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

In the answer box provided, with words, graphs, tables or equations, show your solution to the problem. Only work within the answer box will be scored.

4.

5. Find the coordinates of the image of the point (4,6) when it is reflected across the line y=11. a. (4, -6) c. (4, 16) b. (4, -5) d. (4, 17)

6. Find the translation of the triangle along .

v

a. c.

b. d.

7. Rotate with vertices R(4, –1), S(5, 3), and Q(3, 1) by 90° about the origin. a.

R

S

Q

R'S'

Q'

x

y c.

R

S

QR'

S'

Q'

x

y

b.

R

S

Q

R'S'

Q'

x

y d.

R

S

Q

R'

S'

Q'

x

y

8. State whether the transformation appears to be a translation. Provide specific reasons as to why or why not this transformation is a translation. If it does not appear to be a translation, state the correct transformation.

In the answer box provided, with words, graphs, tables or equations, show your solution to the problem. Only work within the answer box will be scored.

8.

9. On a sketch of a mural, 3 inches represents one foot in the mural. A door in the sketch is 2 inches wide by 5 inches high. What is the perimeter of the door in the mural expressed in inches? a. 4 in. c. 20 in. b. 8 in. d. 56 in. 10. A triangle with vertices , , and is given. Which of the following is the image of the triangle under a dilation with a scale factor of centered at the origin?

A B

C4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

a.

A B

CA'B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y c.

A B

C

A'B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

b.

A B

C

A' B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y d.

A B

C A' B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

Key Student Page

Geometry / Day #10 Coordinate Geometry

1. Ann wants to create a design to decorate her Geometry binder. She reflects part of the design across line p and then reflects the image across line n. Describe a single transformation that moves the part of the design from its starting position to its final position.

Start

Finish

n

p

a. rotation of 180° about the origin (Correct Answer) b. rotation of 90° about the origin c. translation along the line d. reflection across the line

Solution: By theorem, the composition of two reflections across intersecting lines is equivalent to a rotation about the point of intersection, and the angle of rotation is twice the measure of the angle between the lines. Since the lines are perpendicular, they form a 90° angle. The angle of rotation is . Feedback A Correct! B Use the theorem about the compositions of two reflections across two intersecting lines to help

you. C Use the theorem about the compositions of two reflections across two intersecting lines to help

you. D Use the theorem about the compositions of two reflections across two intersecting lines to help

you.

2. Given the rectangle and the center of dilation P, which of the following is a dilation with a scale factor of 2?

P

(Correct Answer) a.

P

c.

P

b.

P

d.

P

Solution: Step 1 Draw a line through P and each vertex. Step 2 On each line mark the point that is twice the distance from P to the vertex. Step 3 Connect the vertices of the image.

P

AB

CD

A'B'

C'D'

Feedback A This rectangle is twice the width, but not twice the height. B This rectangle is the right size, but the dilation is not centered at P. C Correct! D This is a dilation with a scale factor of one half.

3. In a dance performance, four dancers form a diamond with vertices . Then, they move along the dance floor following the translation

vector, . There they pause, and then move again along the same vector. What are their coordinates after six such translations? a. (Correct Answer) b. c. d. Solution: The complete translation is .

Feedback A Correct! B Add 24 to each y-coordinate. C Add 24 to each y-coordinate. D Add 24 to each y-coordinate.

4. Describe the dilation in the graph

A B

C

A' B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

In the answer box provided, with words, graphs, tables or equations, show your solution to the problem. Only work within the answer box will be scored.

. 4. Solution: (1 Point) Noting the coordinates of either A, B, or C and the coordinates of the representative A’, B’, or C’ from the dilated triangle. Such as A (2, 1) and A’ (6, 3) OR B (4, 1) and B’ (12, 3) OR C (4, -3) and C’ (12, -9). (1 Point) It is a dilation of 3 centered at the origin.

5. Find the coordinates of the image of the point (4,6) when it is reflected across the line y=11. a. (4, -6) c. (4, 16) (Correct Answer) b. (4, -5) d. (4, 17) Solution:

(4, 6)

(4, 16)

preimage

image

4 8 12 16 20 24 28–4–8–12 x

4

8

12

16

20

24

28

–4

–8

–12

y

Feedback A This is the point reflected over the x-axis. Find the point reflected over the given line. B Graph the point and the line of symmetry to help you visualize the reflection. C Correct! D The reflected image should have the same distance.

6. Find the translation of the triangle along .

v

(Correct Answer) a. c.

b. d.

Solution: Step 1 Draw a line parallel to the vector through each vertex of the triangle. Step 2 Measure the length of the vector. Then, from each vertex mark off this distance in the same direction as the vector, on each of the parallel lines.

v

v

Step 3 Connect the image of the vertices.

v

Feedback A Correct! B The vector v translates each point to the left three units, then up three units. C The vector v translates each point to the left three units, then up three units. D The vector v translates each point to the left three units, then up three units.

7. Rotate with vertices R(4, –1), S(5, 3), and Q(3, 1) by 90° about the origin. (Correct Answer) a.

R

S

Q

R'S'

Q'

x

y c.

R

S

QR'

S'

Q'

x

y

b.

R

S

Q

R'S'

Q'

x

y d.

R

S

Q

R'

S'

Q'

x

y

Solution: The image of (x, y) is (–y, x). R(4, –1) (1, 4) S(5, 3) (–3, 5) Q(3, 1) (–1, 3) Graph the preimage and the image. Feedback A Correct! B The rotation is 90° counterclockwise, not clockwise. C This is a rotation by 180° about the origin. D This is a reflection across the y-axis.

8. State whether the transformation appears to be a translation. Provide specific reasons as to why or why not this transformation appears to be a translation. If it does not appear to be a translation, state the correct transformation and provide specific reasons that support the correct transformation.

In the answer box provided, with words, graphs, tables or equations, show your solution to the problem. Only work within the answer box will be scored.

8. Solution: This transformation does not appear to be a transformation. A translation requires each point from the original figure to move the same distance and the same direction to the transformed figure. Each point has not moved the same distance and the same direction between the original figure and the transformed figure. (2 points) This transformation appears to be a reflection. A reflection of the original image appears to be flipped across a line. (2 points)

9. On a sketch of a mural, 3 inches represents one foot in the mural. A door in the sketch is 2 inches wide by 5 inches high. What is the perimeter of the door in the mural expressed in inches? a. 4 in. c. 20 in. b. 8 in. d. 56 in. (Correct Answer) Solution: Because 3 inches on the sketch represents 12 inches in the mural, the mural is a dilation of the sketch by a factor of 4. Find the dimensions of the door in the mural. Multiply each dimension by 4.

in. in.

Find the perimeter of the door.

in. Feedback A This is the scale factor for the dilation. Now find the dimensions and the perimeter. B This is the width of the door, now find the height and the perimeter. C This is the height of the door, now find the width and the perimeter. D Correct!

10. A triangle with vertices , , and is given. Which of the following is the image of the triangle under a dilation with a scale factor of centered at the origin?

A B

C4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

(Correct Answer) a.

A B

CA'B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y c.

A B

C

A'B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

b.

A B

C

A' B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y d.

A B

C A' B'

C'

4 8 12–4–8–12 x

4

8

12

–4

–8

–12

y

Solution: The dilation of is

Feedback A Correct! B Multiply the coordinates of each of the vertices by –3. C Multiply the coordinates of each of the vertices by –3. D Multiply the coordinates of each of the vertices by –3.