LESSON 9–6
Dilations
Five-Minute Check (over Lesson 9–5)
TEKS
Then/Now
Key Concept: Dilation
Example 1: Draw a Dilation
Example 2: Real-World Example: Find the Scale of a
Dilation
Key Concept: Dilations in the Coordinate Plane
Example 3: Dilations in the Coordinate Plane
Example 4: Compositions of Dilations
Over Lesson 9–5
State whether the figure
appears to have line symmetry.
If so, how many lines of
symmetry does it have?
A. yes; 4 lines
B. yes; 3 lines
C. yes; 2 lines
D. This figure does not
have line symmetry.
Over Lesson 9–5
State whether the figure
appears to have line symmetry.
If so, how many lines of
symmetry does it have?
A.
yes; 8 lines
B.
yes; 4 lines
C.
yes; 2 lines
D.
This figure does not have line symmetry.
Over Lesson 9–5
The figure has rotational
symmetry. State the order and
magnitude of symmetry.
A. 5; 72°
B. 5; 45°
C. 6; 60°
D. 6; 72°
Over Lesson 9–5
The figure has rotational
symmetry. State the order and
magnitude of symmetry.
A. 8; 60°
B. 8; 45°
C. 10; 45°
D. 10; 36°
Over Lesson 9–5
What is the order and magnitude of symmetry of
a regular hexagon?
A. order 2, magnitude 180°
B. order 3, magnitude 120°
C. order 6, magnitude 60°
D. order 12, magnitude 30°
Targeted TEKS
G.3(A) Describe and perform transformations of figures in a
plane using coordinate notation.
G.3(B) Determine the image or pre-image of a given twodimensional figure under a composition of rigid
transformations, a composition of non-rigid transformations,
and a composition of both, including dilations where
the center can be any point in the plane.
Also addresses G.10(B).
Mathematical Processes
G.1(C), G.1(E)
You identified dilations and verified them as
similarity transformations.
• Draw dilations.
• Draw dilations in the coordinate plane.
Draw a Dilation
Copy trapezoid PQRS and point C. Then use a
ruler to draw the image of trapezoid PQRS under a
dilation with center C and scale factor 3.
Since k > 1, the dilation is an enlargement of trapezoid
PQRS.
Draw a Dilation
Copy trapezoid PQRS and point C. Then use a
ruler to draw the image of trapezoid PQRS under a
dilation with center C and scale factor 3.
Locate P', Q', R', and S' so that
Draw a Dilation
Copy trapezoid PQRS and point C. Then use a
ruler to draw the image of trapezoid PQRS under a
dilation with center C and scale factor 3.
Answer: Draw trapezoid P'Q'R'S'.
Which diagram shows the dilation image of ΔLMN
with center C and
?
A.
B.
C.
D.
Find the Scale Factor of a Dilation
PUPPETS To create the illusion of a “life-sized”
image, puppeteers sometimes use a light source to
show an enlarged image of a puppet projected on a
screen or wall. Suppose that the distance between
a light source L and the puppet is 24 inches (LP). To
what distance PP' should
you place the puppet from
the screen to create a
49.5-inch tall shadow (I'M')
from a 9-inch puppet?
Find the Scale Factor of a Dilation
Analyze
This problem involves a dilation. The
center of the dilation is L, LP = 24 in.,
IM = 9 in., I'M' = 49.5 in. You are
asked to find PP'.
Formulate
Find the scale factor of the dilation
from the preimage IM to the image
I'M'. Use the scale factor to find LP
and then use LP and LP' to find PP'.
Find the Scale Factor of a Dilation
Determine
The scale factor k of the
enlargement is the ratio of the length
on the image to a corresponding
length on the preimage.
Scale factor of image
image = I'M', preimage = IM
Divide.
Find the Scale Factor of a Dilation
Use this scale factor of 5.5 to find LP'.
LP' = k(LP)
Definition of dilation
= 5.5(24)
k = 5.5 and LP = 24
= 132
Multiply.
Use LP' and LP to find PP'.
LP + PP' = LP'
Segment Addition
24 + PP' = 132
LP = 24 and LP' = 132
PP' = 108
Subtract 24 from each side.
Find the Scale Factor of a Dilation
Answer: So, the puppet should be placed so that the
distance from it to the screen (PP') is 108
inches.
Justify Since the dilation is an enlargement, the
scale factor should be greater than 1.
Evaluate Since 5.5 > 1, the scale factor is
reasonable.
PUPPETS Suppose you have a similar situation
with the puppet and light source. The distance
between the light source L and the puppet is 30
inches (LP). To what distance should you place the
puppet from the screen to create a 54-inch tall
shadow (I'M') from a 6-inch puppet?
A. 100 inches
B. 180 inches
C. 220 inches
D. 240 inches
Dilations in the Coordinate Plane
A. Trapezoid EFGH has vertices E(–8, 4), F(–4, 8),
G(8, 4) and H(–4, –8). Graph the image of EFGH
after a dilation centered at the origin with a scale
factor of
Multiply the x- and y-coordinates of each vertex by the
scale factor,
Dilations in the Coordinate Plane
Graph the preimage and image.
Answer: E'(–2, 1), F'(–1, 2), G'(2, 1), H'(–1, –2)
Dilations in the Coordinate Plane
B. Triangle ABC has vertices A(–2, -4), B(3, 2),
C(0, 4). Graph the image of ABC after a dilation
centered at (1, 2) with a scale factor of 1.5.
Dilations in the Coordinate Plane
Graph the preimage and image.
Answer:
Triangle ABC has vertices A(–1, 1), B(2, –2), and
C(–1, –2). Find the image of ΔABC after a dilation
centered at the origin with a scale factor of 2. Sketch
the preimage and the image.
A.
B.
C.
D. none of the above
Compositions of Dilations
LESSON 9–6
Dilations