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Warm-Up Exercises
1. Given RST ~ XYZ with
RS
ST
RT
3
=
=
.
Find
.
XY
YZ
XZ
2
ANSWER
3
2
2. Given EFG ~ MNP with
EF = 2 . If FG = 4.5, find NP.
MN 1
ANSWER
2.25
Warm-Up Exercises
Target
Show that triangles are similar.
You will…
Identify similarity transformations
called dilations.
Vocabulary
Warm-Up Exercises
• dilation – a transformation that preserves angle
measures and proportional side lengths; sometimes
called a similarity transformation
• scale factor of dilation – ratio of side lengths
image : preimage
descriptions also include a “center” (point of dilation)
• dilations and similarity – if a dilation can be used to
move one figure onto another, the two figures are
similar.
• dilations and rigid motions – if a dilation followed by
any combination of rigid motions can be used to
move one figure onto the other, the two figures are
similar.
EXAMPLE
Warm-Up1Exercises
Describe a dilation
FEG is similar to
moves FEG onto
preimage
FDH. Describe the dilation that
FDH.
image
SOLUTION
The figure shows a dilation with center F.
𝟐
The scale factor (image : preimage) is because
𝟏
the ratio of FH : FG is 20 : 10 or 2 : 1.
EXAMPLE
Warm-Up2Exercises
Describe a combination of transformations
ABC is similar to FGE. Describe a combination
of transformations that moves ABC onto FGE.
In the figure:
AB = 9, FG = 6
SOLUTION
image
reduce
(dilation)
preimage
A dilation with center B and scale
image:preimage
2
factor
moves ABC onto DBE
3
Then a rotation of DBE with center E
moves DBE onto FGE. The angle of
rotation is equal to the measure of C.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
The two figures are similar. Describe the
transformation(s) that move the blue figure onto the red
figure.
1.
2.
ANSWER
ANSWER
dilation with center
B and scale factor 7
3
dilation with center D, scale
factor 2 and reflection
3
EXAMPLE
Warm-Up3Exercises
Use transformations to show figures are not similar
Use transformations to explain why ABCDE and KLQRP
are not similar. (The red polygon is the image.)
The scale factor of the dilation with center A is 2 : 3.
However, the angle measures have not been preserved
with angles P, Q and R.
EXAMPLE
Warm-Up3Exercises
Use transformations to show figures are not similar
SOLUTION
Corresponding sides in the pentagons 2
are proportional with a scale factor of 3 .
However, this does not necessarily
mean the pentagons are similar.
A dilation with center A and scale
factor 2 moves ABCDE onto
3
AFGHJ. Then a reflection moves
AFGHJ onto KLMNP.
KLMNP does not exactly coincide with KLQRP, because
not all of the corresponding angles are congruent.
(Only
A and K are congruent.) Since angle measure
is not preserved, the two pentagons are not similar.
EXAMPLE
Warm-Up4Exercises
Use similar figures
GRAPHIC DESIGN: a design for a party mask is made
using all equilateral triangles and a scale factor of 1 .
2
a. Describe transformations that move triangle A onto
triangle B.
b. Describe why triangles C and D are similar by using
the given information.
EXAMPLE
Warm-Up4Exercises
Use similar figures
Describe the
transformations that
move triangle A onto
triangle B.
SOLUTION
a. The figure shows a dilation
with scale factor 1 .
2
followed by a clockwise
rotation of 60°.
EXAMPLE
Warm-Up4Exercises
Use similar figures
Describe why triangles
C and D are similar by
using the given
information.
SOLUTION
b. Triangles C and D are similar
because all pairs of
corresponding sides are
1
proportional with a ratio of
2
and all pairs of corresponding
angles of equilateral triangles
have the same measure.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Refer to the floor tile designs shown below. In each
design, the red shape is a regular hexagon.
Tile design 1 is made using two hexagons. Explain
why the red and blue hexagons are not similar.
The red hexagon has all sides congruent, but the blue
hexagon has 3 shorter sides and 3 longer sides, so
ratios of corresponding side lengths are not constant.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Tile design 2 is made using two similar geometric
shapes. Describe the transformations that move the
blue hexagon to the red hexagon.
dilation followed by a rotation of 30° about the
center of the figures
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Tile design 3 shows congruent angles and sides.
Explain why the red and blue hexagons are similar,
using the given information.
All angles are congruent, so angle measure is
preserved, and all side lengths are congruent in each
hexagon, so the ratio of any two corresponding side
lengths is constant.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
If the lengths of all the sides of one polygon are
proportional to the lengths of all the corresponding
sides of another polygon, must the polygons be
similar? Explain.
No; even though corresponding sides might be
proportional, if corresponding angles are not
congruent, the polygons are not similar.