Teacher Page
Geometry / Day # 10
9-12.G.3.1
9-12.G.3.2
9-12.G.3.3
9-12.G.3.4
Coordinate Geometry (45 min.)
Use rigid motions (compositions of reflections, translations and rotations) to
determine whether two geometric figures are congruent in a coordinate plane.
Sketch a planar figure that is the result of given transformations (i.e. translation,
reflection, rotation, and/or dilation)
Identify similarity in terms of transformations
Determine the effects of transformations on linear and area measurements of the
original planar figure.
1. Reflect a figure with vertices
coordinates of the new image.
across the x-axis. Find the
a.
b.
c.
d.
vertices A(3, 1), B(4, 5), and C(2, 3). Rotate
across the x-axis.
y
a.
c.
90° about the origin and then reflect it
2.
y
B
B"
B
B'
A'
C
C"
C'
A
C
A"
x
A
x
A'
C"
A"
C'
B"
B'
b.
d.
y
y
B
B
C
C
A
A
x
C"
C'
A" A'
B"
A"
B'
B"
x
A'
C"
C'
B'
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.
y
10
5
–10
–5
5
10
x
–5
–10
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
1
centered at the origin.
2
B. Find the coordinates
y
10
5
–10
–5
5
10
x
–5
–10
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.)
across the x-axis.
1. Reflect a figure with vertices
Find the coordinates of the new image.
a.
b.
c.
d.
(Correct Answer)
Solution:
The reflection of
A
B
C
D
across the x-axis is
.
Feedback
The reflection of (x, y) on the x-axis is (x, –y).
Correct!
The reflection of (x, y) on the x-axis is (x, –y).
The reflection of (x, y) on the x-axis is (x, –y).
vertices A(3, 1), B(4, 5), and C(2, 3). Rotate
2.
the x-axis.
90° about the origin and then reflect it across
(Correct Answer)
a.
c.
y
y
B
B"
B
B'
A'
C
C"
C'
A
C
A"
x
A
x
A'
C"
A"
C'
B"
B'
b.
d.
y
y
B
B
C
C
A
A
x
C"
C'
C"
A" A'
B"
A"
B'
B"
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)
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)
A
B
C
D
x
A'
C'
B'
(–3, 2).
(–3, –2).
Feedback
Correct!
This is a rotation of 90° clockwise about the origin and a reflection is across the y-axis.
This is a rotation of 180° about the origin, and then a reflection across the x-axis.
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.
y
10
5
(3, 4)
–10 (–9, –1)
–5
5
–5
10
(10, –5)
–10
Solution:
Original Triangle Sketched (1 Point)
Translated triangle vertices (1 Point)
(3, 11)
(10, 2)
(-9, 6)
x
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
.
1
centered at the origin. (1 Point)
2
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.
p
Start
n
Finish
a.
b.
rotation of 180° about the origin
rotation of 90° about the origin
c.
d.
translation along the line
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
c.
a.
P
P
b.
d.
P
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.
y
12
8
A'
4
A
–12
–8
–4
4
–4
B'
B
8
12
x
C
–8
C'
–12
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.
b.
(4, -6)
(4, -5)
c.
d.
6. Find the translation of the triangle along
v
a.
c.
b.
d.
(4, 16)
(4, 17)
.
7. Rotate
a.
with vertices R(4, –1), S(5, 3), and Q(3, 1) by 90° about the origin.
y
y
c.
S'
R'
S
S
Q'
Q
R'
Q
x
R
x
R
Q'
S'
b.
d.
y
S
y
S'
S
Q
Q'
R
Q'
R'
S'
x
R'
Q
R
x
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.
b.
4 in.
8 in.
c.
d.
20 in.
56 in.
,
, and
10. A triangle with vertices
triangle under a dilation with a scale factor of
is given. Which of the following is the image of the
centered at the origin?
y
12
8
4
A
–12
–8
–4
B
4
8
12
x
C
–4
–8
–12
a.
c.
y
y
12
12
8
8
C'
B'
4
A
–12
–8
B'
–4
A'
A'
4
B
A
4
8
12
x
–12
–8
–4
C
–4
B
4
–4
–8
8
12
x
C
–8
C'
–12
b.
–12
d.
y
y
12
12
C'
8
8
A'
4
A
–12
–8
–4
4
B
4
–4
B'
A
8
12
C
x
–12
–8
–4
4
–4
–8
–8
C'
–12
B
–12
C
8
A'
12
B'
x
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.
p
Start
n
Finish
a.
b.
c.
d.
rotation of 180° about the origin (Correct Answer)
rotation of 90° about the origin
translation along the line
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
A
B
C
D
.
Feedback
Correct!
Use the theorem about the compositions of two reflections across two intersecting lines to help
you.
Use the theorem about the compositions of two reflections across two intersecting lines to help
you.
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)
c.
a.
P
P
d.
b.
P
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.
A'
B'
A
B
P
D
C
D'
C'
A
B
C
D
Feedback
This rectangle is twice the width, but not twice the height.
This rectangle is the right size, but the dilation is not centered at P.
Correct!
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?
(Correct Answer)
a.
b.
c.
d.
Solution:
.
The complete translation is
A
B
C
D
Feedback
Correct!
Add 24 to each y-coordinate.
Add 24 to each y-coordinate.
Add 24 to each y-coordinate.
4. Describe the dilation in the graph
y
12
8
A'
4
A
–12
–8
–4
B
4
–4
B'
8
12
x
C
–8
C'
–12
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.
b.
(4, -6)
(4, -5)
c.
d.
(4, 16) (Correct Answer)
(4, 17)
Solution:
y
28
24
20
16 16)
(4,
image
12
8
preimage
(4, 6)
4
–12 –8
–4
–4
4
8
12
16
20
24
28
x
–8
–12
A
B
C
D
Feedback
This is the point reflected over the x-axis. Find the point reflected over the given line.
Graph the point and the line of symmetry to help you visualize the reflection.
Correct!
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
A
B
C
D
Feedback
Correct!
The vector v translates each point to the left three units, then up three units.
The vector v translates each point to the left three units, then up three units.
The vector v translates each point to the left three units, then up three units.
with vertices R(4, –1), S(5, 3), and Q(3, 1) by 90° about the origin.
7. Rotate
(Correct Answer)
a.
c.
y
y
S'
R'
S
S
Q'
Q
R'
Q
x
R
x
R
Q'
S'
b.
d.
y
S
y
S'
S
Q
Q'
R
x
Q'
R'
S'
Solution:
The image of (x, y) is (–y, x).
(1, 4)
R(4, –1)
S(5, 3)
(–3, 5)
Q(3, 1)
(–1, 3)
Graph the preimage and the image.
A
B
C
D
Feedback
Correct!
The rotation is 90° counterclockwise, not clockwise.
This is a rotation by 180° about the origin.
This is a reflection across the y-axis.
R'
Q
R
x
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.
A
B
C
D
Feedback
This is the scale factor for the dilation. Now find the dimensions and the perimeter.
This is the width of the door, now find the height and the perimeter.
This is the height of the door, now find the width and the perimeter.
Correct!
10. A triangle with vertices
,
, and
triangle under a dilation with a scale factor of
is given. Which of the following is the image of the
centered at the origin?
y
12
8
4
A
–12
–8
–4
B
4
8
x
12
C
–4
–8
–12
(Correct Answer)
a.
c.
y
y
12
12
8
8
C'
B'
4
A
–12
–8
B'
–4
A'
A'
4
B
A
4
8
12
x
–12
–8
–4
C
–4
B
4
–4
–8
8
12
x
C
–8
C'
–12
b.
–12
d.
y
y
12
12
8
8
C'
A'
4
A
–12
–8
–4
4
–4
B'
4
B
A
8
12
C
x
–12
–8
–4
4
–4
–8
–8
C'
–12
B
–12
C
8
A'
12
B'
x
Solution:
The dilation of
A
B
C
D
is
Feedback
Correct!
Multiply the coordinates of each of the vertices by –3.
Multiply the coordinates of each of the vertices by –3.
Multiply the coordinates of each of the vertices by –3.