Transcript Chapter 9 PPT

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Lesson 9-1 Reflections
Lesson 9-2 Translations
Lesson 9-3 Rotations
Lesson 9-4 Tessellations
Lesson 9-5 Dilations
Lesson 9-6 Vectors
Lesson 9-7 Transformations with Matrices
Example 1 Reflecting a Figure in a Line
Example 2 Reflection in the x-axis
Example 3 Reflection in the y-axis
Example 4 Reflection in the Origin
Example 5 Reflection in the Line y = x
Example 6 Use Reflections
Example 7 Draw Lines of Symmetry
Draw the reflected image of quadrilateral WXYZ
in line p.
Step 1 Draw segments
perpendicular to line p from
each point W, X, Y, and Z.
Step 2 Locate W', X', Y',
and Z' so that line p is the
perpendicular bisector of
Points W', X', Y', and Z' are
the respective images of W,
X, Y, and Z.
Step 3 Connect vertices
W', X', Y', and Z'.
Answer: Since points W',
X', Y', and Z' are the images
of points W, X, Y, and Z
under reflection in line p, then
quadrilateral W'X'Y'Z' is the
reflection of quadrilateral
WXYZ in line p.
Draw the reflected image of quadrilateral ABCD in line n.
Answer:
COORDINATE GEOMETRY Quadrilateral ABCD has
vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph
ABCD and its image under reflection in the x-axis.
Compare the coordinates of each vertex with the
coordinates of its image.
Use the vertical grid lines to find the corresponding point
for each vertex so that the x-axis is equidistant from each
vertex and its image.
D'
A(1, 1)  A' (1, –1)
C'
B(3, 2)  B' (3, –2)
C(4, –1)  C' (4, 1)
D(2, –3)  D' (2, 3)
A'
B'
Plot the reflected vertices and connect to form the image
A'B'C'D'.
Answer: The x-coordinates stay the same, but the
y-coordinates are opposite.
That is, (a, b)  (a, –b).
COORDINATE GEOMETRY Quadrilateral LMNP has
vertices L(–1, 1), M(5, 1), N(4, –1), and P(0, –1). Graph
LMPN and its image under reflection in the x-axis.
Compare the coordinates of each vertex with the
coordinates of its image.
Answer:
L(–1, 1)  L' (–1, –1)
M(5, 1)  M' (5, –1)
N(4, –1)  N' (4, 1)
P(0, –1)  P' (0, 1)
The x-coordinates stay the
same, but the y-coordinates
are opposite.
COORDINATE GEOMETRY Quadrilateral ABCD has
vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph
ABCD and its image under reflection in the y-axis.
Compare the coordinates of each vertex with the
coordinates of its image.
Use the horizontal grid lines to find the corresponding point
for each vertex so that the y-axis is equidistant from each
vertex and its image.
B'
A(1, 1)  A' (–1, 1)
A'
B(3, 2)  B' (–3, 2)
C(4, –1)  C' (–4, –1)
D(2, –3)  D' (–2, –3)
C'
D'
Plot the reflected vertices and connect to form the image
A'B'C'D'. The x-coordinates are opposite, but the
y-coordinates stay the same. That is, (a, b)  (–a, b).
Answer: The x-coordinates are opposite, but the
y-coordinates stay the same.
That is, (a, b)  (–a, b).
COORDINATE GEOMETRY Quadrilateral LMNP has
vertices L(–1, 1), M(5, 1), N(4, –1), and P(0, –1). Graph
LMPN and its image under reflection in the y-axis.
Compare the coordinates of each vertex with the
coordinates of its image.
Answer:
L(–1, 1)  L' (1, 1)
M(5, 1)  M' (–5, 1)
N(4, –1)  N' (–4, –1)
P(0, –1)  P' (0, –1)
The x-coordinates are opposite, but the
y-coordinates stay the same.
COORDINATE GEOMETRY Suppose quadrilateral ABCD
with A(1, 2), B(3, 5), C(4, –3), and D(2, –5) is reflected in
the origin. Graph ABCD and its image under reflection
in the origin. Compare the coordinates of each vertex
with the coordinates of its image.
Use the horizontal and vertical distances from each vertex
to the origin to find the coordinates of its image. From A to
the origin is 2 units down and 1 unit left. A' is located by
repeating that pattern from the origin.
A(1, 2)  A' (–1, –2)
B(3, 5)  B' (–3, –5)
D'
C'
C(4, –3)  C' (–4, 3)
D(2, –5)  D' (–2, 5)
Plot the reflected vertices
and connect to form the
image A'B'C'D'. Comparing
coordinates shows that
(a, b)  (–a, –b).
A'
B'
Answer: Both the x-coordinates and y-coordinates are
opposite. That is, (a, b)  (–a, –b).
COORDINATE GEOMETRY Quadrilateral LMNP has
vertices L(–1, 1), M(5, 1), N(4, –1), and P(0, –1). Graph
LMPN and its image under reflection in the origin.
Compare the coordinates of each vertex with the
coordinates of its image.
Answer:
L(–1, 1)  L' (1, –1)
M(5, 1)  M' (–5, –1)
N(4, –1)  N' (–4, 1)
P(0, –1)  P' (0, 1)
Both the x-coordinates
and y-coordinates are
opposite.
COORDINATE GEOMETRY Suppose quadrilateral ABCD
with A(1, 2), B(3, 5), C(4, –3), and D(2, –5) is reflected in
the line
Graph ABCD and its image under
reflection in the line
Compare the coordinates of
each vertex with the coordinates of its image.
The slope of
is perpendicular to
slope is –1. From A to the line
right
unit. From the line
unit to A'.
move down
move down
so its
unit and
unit, right
A(1, 2)  A'(2, 1)
C'
B(3, 5)  B'(5, 3)
C(4, –3)  C'(–3, 4)
D(2, –5)  D'(–5, 2)
B'
D'
A'
Plot the reflected vertices
and connect to form the
image A'B'C'D'.
Answer: The x-coordinate becomes the y-coordinate and
the y-coordinate becomes the x-coordinate.
That is, (a, b)  (b, a).
COORDINATE GEOMETRY Quadrilateral LMNP has
vertices L(–1, 1), M(5, 1), N(4, –1), and P(0, –1). Graph
LMNP and its image under reflection in the line
Compare the coordinates of each vertex with the
coordinates of its image.
Answer:
L(–1, 1)  L' (1, –1)
M(5, 1)  M' (1, 5)
N(4, –1)  N' (–1, 4)
P(0, –1)  P' (–1, 0)
The x-coordinate becomes
the y-coordinate and the
y-coordinate becomes the
x-coordinate.
TABLE TENNIS During a game of table tennis, Dipa
decides that she wants to hit the ball so that it strikes
her side of the table and then just clears the net.
Describe how she should hit the ball using reflections.
Answer: She should mentally reflect the desired position
of the ball in the line of the table and aim toward
the reflected image under the table.
BILLARDS
Dave challenged
Juan to hit the 8
ball in the left
corner pocket.
Describe how Juan
should hit the ball
using reflections.
Answer:
Juan should
mentally reflect the
left corner pocket in
the line that
contains the right
side of the table. If
he hits the ball at
the reflected image
of the pocket, the
ball will strike the
right side and
rebound on a path
toward the left
corner pocket.
Determine how many lines of symmetry a regular
pentagon has. Then determine whether a regular
pentagon has a point of symmetry.
A regular pentagon has five lines of symmetry.
A point of symmetry is a point that is a common point of
reflection for all points on the figure. There is not one point
of symmetry in a regular pentagon.
Answer: 5; no
Determine how many lines of symmetry an equilateral
triangle has. Then determine whether an equilateral
triangle has a point of symmetry.
Answer: 3; no
Example 1 Translations in the Coordinate Plane
Example 2 Repeated Translations
Example 3 Find a Translation Using Reflections
COORDINATE GEOMETRY Parallelogram TUVW has
vertices T(–1, 4), U(2, 5), V(4, 3), and W(1, 2). Graph
TUVW and its image for the translation
(x, y)  (x – 4, y – 5).
This translation moved every point of the preimage
4 units left and 5 units down.
T(–1, 4)
U(2, 5)
V(4, 3)
W(1, 2)




T' (–1 – 4, 4 – 5)
U' (2 – 4, 5 – 5)
V' (4 – 4, 3 – 5)
W' (1 – 4, 2 – 5)
or
or
or
or
Plot the translated vertices and connect to form
parallelogram T'U'V'W'.
T' (–5, –1)
U' (–2, 0)
V' (0, –2)
W' (–3, –3)
Answer:
COORDINATE GEOMETRY Parallelogram LMNP has
vertices L(–1, 2), M(1, 4), N(3, 2), and P(1, 0). Graph
LMNP and its image for the translation
(x, y)  (x + 3, y – 4).
Answer:
ANIMATION The graph shows repeated translations
that result in the animation of a raindrop. Find the
translation that moves raindrop 2 to raindrop 3 and
then the translation that moves raindrop 3 to raindrop 4.
To find the translation from raindrop 2 to raindrop 3, use the
coordinates at the top of each raindrop. Use the coordinates
(1, 2) and (–1, –1) in the formula.
Subtract 1 from each side.
Subtract 2 from each side.
The translation is (x – 2, y – 3) from raindrop 2 to raindrop 3.
Use the coordinates (–1, –1) and (–1, –4) to find the
translation from raindrop 3 to raindrop 4.
(x, y)
(–1, –1)
(–1, –4)
Add 1 to each side.
Add 1 to each side.
The translation is
Answer:
from raindrop 3 to raindrop 4.
ANIMATION The graph shows repeated translations
that result in the animation of a lightning bolt. Find
the translation that moves lightning bolt 3 to lightning
bolt 4 and then the translation that moves lightning
bolt 2 to lightning bolt 1.
Answer: (x – 3, y – 1); (x + 2, y + 2)
In the figure, lines p and q are parallel. Determine
whether the pink figure is a translation image of the
blue preimage, quadrilateral EFGH.
Reflect quadrilateral EFGH in line p. The result is the green
image, quadrilateral E'F'G'H'. This is not a reflection of
quadrilateral EFGH in line p, so quadrilateral E'F'G'H' is
not a translation of quadrilateral EFGH.
Answer: Quadrilateral E''F''G''H'' is not a translation
image of quadrilateral EFGH.
In the figure, lines n and m are parallel. Determine
whether A''B''C'' is a translation image of the
preimage, ABC.
Answer: Yes, A''B''C'' is the translation image of ABC.
Example 1 Draw a Rotation
Example 2 Reflections in Intersecting Lines
Example 3 Identifying Rotational Symmetry
Triangle DEF has vertices D(–2, –1), E(–1, 1), and
F(1, –1). Draw the image of DEF under a rotation of
115° clockwise about the point G(–4, –2).
First draw DEF and plot point G.
E
Draw a segment from point G to
point D.
Use a protractor to measure a 115°
angle clockwise with
as one side.
Draw
Use a compass to copy
onto
Name the segment
Repeat with points E and F.
D
F
G
115
D'
E'
F'
R
D'E'F' is the image of DEF under a 115° clockwise
rotation about point G.
Answer:
E
D
F
D'
E'
F'
Triangle ABC has vertices A(1, –2), B(4, –6), and
C(1, –6). Draw the image of ABC under a rotation of
70° counterclockwise about the point M(–1, –1).
Answer:
Find the image of parallelogram WXYZ under
reflections in line p and then line q.
First reflect parallelogram
WXYZ in line p. Then label the
image W'X'Y'Z'.
Next, reflect the image in line q.
Then label the image W''X''Y''Z''.
Answer:
Parallelogram W''X''Y''Z'' is the
image of parallelogram WXYZ
under reflections in line p and q.
Find the image of ABC under reflections in line m and
then line n.
Answer:
QUILTS Use the quilt by Judy Mathieson shown below.
Identify the order and magnitude of the symmetry in
the medium star directly to the left of the large star in
the center of the quilt.
Answer: The medium star
directly to the left of the
large star in the center
has rotational symmetry
of order 16 and a
magnitude of 22.5°.
QUILTS Use the quilt by Judy Mathieson shown below.
Identify the order and magnitude of the symmetry in
the tiny star above the medium-sized star in
Example 3a.
Answer: The tiny star has
rotational symmetry of
order 8 and magnitude
of 45°.
QUILTS Use the quilt by Judy Mathieson shown below.
Identify the order and magnitude of the symmetry in
each part of the quilt.
a. star in the upper left corner
Answer: 8; 45°
b. medium-sized star directly
in center of quilt
Answer: 20; 18°
Example 1 Regular Polygons
Example 2 Semi-Regular Tessellation
Example 3 Classify Tessellations
Determine whether a regular 16-gon tessellates the
plane. Explain.
Let 1 represent one interior angle of a regular 16-gon.
m1
Interior Angle Theorem
Substitution
Simplify.
Answer: Since 157.5 is not a factor of 360,
a 16-gon will not tessellate the plane.
Determine whether a regular 20-gon tessellates the
plane. Explain.
Answer: No; 162 is not a factor of 360.
Determine whether a semi-regular tessellation can be
created from regular nonagons and squares, all having
sides 1 unit long.
Solve algebraically.
Each interior angle of a regular nonagon measures
or 140°.
Each angle of a square measures 90°. Find whole-number
values for n and s such that
All whole numbers greater than 3 will result in a negative
value for s.
Substitution
Simplify.
Subtract from
each side.
Divide each side
by 90.
Answer: There are no whole number values
for n and s so that
Determine whether a semi-regular tessellation can be
created from regular hexagon and squares, all having
sides 1 unit long. Explain.
Answer: No; there are no whole number values for h and s
such that
STAINED GLASS Stained glass is a very popular design
selection for church and cathedral windows. It is also
fashionable to use stained glass for lampshades,
decorative clocks, and residential windows. Determine
whether the pattern is a tessellation. If so, describe it
as uniform, regular, semi-regular, or not uniform.
Answer: The pattern is a tessellation because at the
different vertices the sum of the angles is 360°.
The tessellation is not uniform because each
vertex does not have the same arrangement of
shapes and angles.
STAINED GLASS Stained glass is a very popular design
selection for church and cathedral windows. It is also
fashionable to use stained glass for lampshades,
decorative clocks, and residential windows. Determine
whether the pattern is a tessellation. If so, describe it
as uniform, regular, semi-regular, or not uniform.
Answer: tessellation, not uniform
Example 1 Determine Measures Under Dilations
Example 2 Draw a Dilation
Example 3 Dilations in the Coordinate Plane
Example 4 Identify Scale Factor
Example 5 Scale Drawing
Find the measure of the dilation image or the preimage
of
using the given scale factor of
.
Dilation Theorem
Multiply.
Answer: 45
Find the measure of the dilation image or the preimage
of
using the given scale factor
.
Dilation Theorem
Multiply each side by
Answer: 10.5
Find the measure of the dilation image or the preimage
of
using the given scale factor.
a.
Answer: 32
b.
Answer: 36
Draw the dilation image of trapezoid PQRS with center
C and
Since
the dilation is an enlargement of trapezoid
PQRS.
Draw
and S' will lie on
. Since r is negative, P', Q', R',
respectively.
Draw the dilation image of trapezoid PQRS with center
C and
Since
the dilation is an enlargement of trapezoid
R'
PQRS.
S'
P'
Q'
Locate P', Q', R', and S' so that
Draw the dilation image of trapezoid PQRS with center
C and
Since
the dilation is an enlargement of trapezoid
R'
PQRS.
S'
P'
Q'
Answer: Draw trapezoid P'Q'R'S'.
Draw the dilation image of LMN with center C
and
Answer:
COORDINATE GEOMETRY Trapezoid EFGH has
vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8). Find the
image of trapezoid EFGH after a dilation centered at the
origin with a scale factor of
Sketch the preimage and
the image.
Preimage (x, y)
Image
E(–8, 4)
F(–4, 8)
E'(–2, 1)
F'(–1, 2)
G(8, 4)
H(–4, –8)
G'(2, 1)
H'(–1, –2)
Answer: E'(–2, 1), F'(–1, 2), G'(2, 1), H'(–1, –2)
COORDINATE GEOMETRY Triangle ABC has vertices
A(–1, 1), B(2, –2), and C(–1, –2). Find the image of ABC
after a dilation centered at the origin with a scale factor
of 2. Sketch the preimage and the image.
Answer: A'(–2, 2), B'(4, –4), C' (–2, –4)
Determine the scale factor used for the dilation
with center C. Determine whether the dilation is an
enlargement, reduction, or congruence transformation.
image length
preimage length
Simplify.
Since the scale factor is less than 1, the dilation is
a reduction.
Answer:
; reduction
Determine the scale factor used for the dilation
with center C. Determine whether the dilation is an
enlargement, reduction, or congruence transformation.
image length
preimage length
Simplify.
Since the image falls on the opposite side of the center, C,
than the preimage, the scale factor is negative. So the
scale factor is –1. The absolute value of the scale factor
equals 1, so the dilation is a congruence transformation.
Answer: –1; congruence transformation
Determine the scale factor used for each dilation with
center C. Determine whether the dilation is an
enlargement, reduction, or congruence transformation.
a.
Answer:
reduction
Determine the scale factor used for each dilation with
center C. Determine whether the dilation is an
enlargement, reduction, or congruence transformation.
b.
Answer: 2; enlargement
MULTIPLE-CHOICE TEST ITEM
Sharetta built a frame for a photograph that is 20
centimeters by 25 centimeters. The frame measures
400 millimeters by 500 millimeters. Which scale factor
did she use?
A 2
B 3
C
D
Read the Test Item
The photograph’s dimensions are given in centimeters, and
the frame’s dimensions are in millimeters. You need to
convert from millimeters to centimeters in the problem.
Solve the Test Item
Step 1
Convert from millimeters to centimeters.
or 40 centimeters
or 20 centimeters
Step 2
Find the scale factor.
frame length
photo length
Simplify.
Step 3
Sharetta used a scale factor of 2 to build the
frame. Choice A is the correct answer.
Answer: A
MULTIPLE-CHOICE TEST ITEM
Ruben is making a scale drawing of the front of his
house. His house is 48 feet wide and 30 feet high at its
highest point. Ruben decides on a dilation reduction
factor of
What size poster board will he need to
make a complete drawing?
A 19 in. by 26 in.
B 22 in. by 30 in.
C 20.5 in. by 28 in.
D 16 in. by 29 in.
Answer: B
Example 1 Write Vectors in Component Form
Example 2 Magnitude and Direction of a Vector
Example 3 Translations with Vectors
Example 4 Add Vectors
Example 5 Solve Problems Using Vectors
Write the component form of
Find the change of x values and the corresponding
change in y values.
Component form of vector
Simplify.
Answer: Because the magnitude and direction of a vector
are not changed by translation, the vector
represents the same vector as
Write the component form of
Answer:
Find the magnitude and direction of
and T(4, –7).
for S(–3, –2)
Find the magnitude.
Distance Formula
Simplify.
Use a calculator.
Graph
to determine how to find the direction.
Draw a right triangle that has
as its hypotenuse
and an acute angle at S.
tan S
Substitution
Simplify.
Use a calculator.
A vector in standard position that is equal to
forms a
–35.5° degree angle with the positive x-axis in the fourth
quadrant. So it forms a
angle with
the positive x-axis.
Answer:
has a magnitude of about 8.6 units and a
direction of about 324.5°.
Find the magnitude and direction of
and B(–2, 1).
Answer:  5.7; 225°
for A(2, 5)
Graph the image of quadrilateral HJLK with vertices
H(–4, 4), J(–2, 4), L(–1, 2) and K(–3, 1) under the
translation of v
Answer:
First graph quadrilateral HJLK.
Next translate each
vertex by , 5 units right
and 5 units down.
Connect the vertices for
quadrilateral
.
Graph the image of triangle ABC with vertices A(7, 6),
B(6, 2), and C(2, 3) under the translation of v
Answer:
Graph the image of EFG with vertices E(1, –3), F(3, –1),
and G(4, –4) under the translation a
and b
Graph EFG.
F'
Method 1 Translate two times.
Translate EFG by a. Then
translate EFG by b.
E'
G'
F
Translate each vertex 4 units
left and 2 units up.
Then translate each vertex
of E'F'G' 2 units right and
3 units up. Label the image
E'F'G'.
E
G
Method 2 Find the resultant, and then translate.
Add a and b.
F'
E'
G'
Translate each vertex 2
units left and 5 units up.
Answer: Notice that the vertices
for the image are the
same for either method.
F
E
G
Graph the image of ABC with vertices A(0, 6), B(–1, 2),
and C(–5, 3) under the translation by m
and
n
Answer:
CANOEING Suppose a person is canoeing due east
across a river at 4 miles per hour. If the river is flowing
south at 3 miles an hour, what is the resultant direction
and velocity of the canoe?
The initial path of the canoe is due east, so a vector
representing the path lies on the positive x-axis 4 units
long. The river is flowing south, so a vector representing
the river will be parallel to the negative y-axis 3 units long.
The resultant path can be represented by a vector from
the initial point of the vector representing the canoe to the
terminal point of the vector representing the river.
Use the Pythagorean Theorem.
Pythagorean Theorem
Simplify.
Take the square root of each side.
The resultant velocity of the canoe is 5 miles per hour.
Use the tangent ratio to find the direction of the canoe.
Use a calculator.
The resultant direction of the canoe is about 36.9°
south of due east.
Answer: Therefore, the resultant vector is 5 miles per hour
at 36.9° south of due east.
CANOEING Suppose a person is canoeing due east
across a river at 4 miles per hour. If the current reduces
to half of its original speed, what is the resultant
direction and velocity of the canoe?
Use scalar multiplication to find the magnitude of the vector
for the river.
Magnitude of
Simplify.
Next, use the Pythagorean Theorem to find the magnitude
of the resultant vector.
Pythagorean Theorem
Simplify.
Take the square root of each side.
Then, use the tangent ratio to find the direction of the canoe.
Use a calculator.
Answer: If the current reduces to half its original speed,
the canoe travels along a path approximately 20.6°
south of due east at about 4.3 miles per hour.
KAYAKING Suppose a person is kayaking due east
across a lake at 7 miles per hour.
a. If the lake is flowing south at 4 miles an hour, what is
the resultant direction and velocity of the canoe?
Answer: Resultant direction is about 29.7° south of due
east; resultant velocity is about 8.1 miles per hour.
b. If the current doubles its original speed, what is the
resultant direction and velocity of the kayak?
Answer: Resultant direction is about 48.8° south of due
east; resultant velocity is about 10.6 miles
per hour.
Example 1 Translate a Figure
Example 2 Dilate a Figure
Example 3 Reflections
Example 4 Use Rotations
Use a matrix to find the coordinates of the vertices
of the image of quadrilateral EFGH with E(5, 3),
F(2, –6), G(–5, –5), and H(–3, 4) under the translation
To translate the figure 3 units left, add –3 to each
x-coordinate. To translate the figure 6 units up, add 6
to each y-coordinate. This can be done by adding the
translation matrix to the vertex matrix of quadrilateral EFGH.
Vertex Matrix of
Translation
Vertex Matrix of
quadrilateral EFGH
Matrix
quadrilateral E'F'G'H'
Answer: The coordinates of quadrilateral E'F'G'H'
are E' (2, 9), F' (–1, 0), G' (–8, 1), and H' (–6, 10).
Use a matrix to find the coordinates of the vertices of
the image of triangle ABC with A(–1, 3), B(3, 6), and
C(6, –5) under a translation
Answer: A'(1, 0), B'(5, 3), C'(8, –8)
Triangle ABC has vertices A(4, –8), B(–12, –8), and
C(–24, 12). Use scalar multiplication to dilate ABC
centered at the origin so that its perimeter is
the
original perimeter.
If the perimeter of a figure is
the original perimeter, then
the lengths of the sides of the figure will be
the measures
of the original lengths. Multiply the vertex matrix by a scale
factor of
.
Answer: The coordinates of the vertices of A'B'C' are
A'(1, –2), B'(–3, –2), and C'(–6, 3).
Triangle EFG has vertices E(2, 8), F(–4, 6), and
G(–2, –10). Use scalar multiplication to dilate EFG
centered at the origin so that its perimeter is 2 times
the original perimeter.
Answer: E'(4, 16),
F'(–8, 12),
G'(–4, –20)
Use a matrix to find the coordinates of the vertices
of the image of
with F(–3, 1) and G(0, 4) after a
reflection in the y-axis.
Write the ordered pairs as a vertex matrix. Then multiply
the vertex matrix by the reflection matrix for the y-axis.
Answer: The coordinates of the vertices of
F'(3, 1) and G'(0, 4). Graph
and
are
Use a matrix to find the coordinates of the vertices
of the image of
with L(4, –4) and M(2, 5) after a
reflection in the x-axis.
Answer: L'(4, 4) and M'(2, –5)
Triangle ABC has vertices A(3, 1), B(5, 2), and C(3, 4).
Use a matrix to find the coordinates of the image under
a 270° counterclockwise rotation about the origin.
Write the vertex matrix for ABC.
Enter the rotation matrix in your calculator as matrix A
and enter the vertex matrix as matrix B. Then multiply.
KEYSTROKES: 2nd [MATRX] 1 2nd [MATRX] 2 Enter
Answer: The coordinates of the vertices of the figure are
A'(1, –3), B'(2, –5), and C'(4, –3).
Triangle ABC has vertices A(3, 1), B(5, 2), and C(3, 4).
What are the coordinates of the image if ABC is
reflected in the y-axis before a 90° counterclockwise
rotation about the origin?
Enter the reflection matrix as matrix C. A reflection is
performed before the rotation. Multiply the reflection
matrix and the vertex matrix.
KEYSTROKES: 2nd [MATRX] 3 2nd [MATRX] 2 Enter
Enter the rotation matrix for 90° as matrix D. Then multiply
the rotation matrix by the resulting matrix CB.
KEYSTROKES: 2nd [MATRX] 4 2nd
Enter
Answer: A'(–1, –3), B'(–2, –5), and C'(–4, –3)
Triangle FGH has vertices F(1, 3), G(–2, 5), and H(4, –7).
a. Use a matrix to find the coordinates of the image under
a 180° counterclockwise rotation about the origin.
Answer: F'(–1, –3), G'(2, –5), and H'(–4, 7)
b. What are the coordinates of the image if FGH is
reflected in the x-axis before a 90° counterclockwise
rotation about the origin?
Answer: F'(3, 1), G'(5, –2), and H'(–7, 4)
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