Download Copyright © by Holt, Rinehart and Winston

Dilations and Similarity in the Coordinate Plane
A dilation is a transformation that changes the size of a figure but not its shape.
The preimage and image are always similar. A scale factor describes how much
a figure is enlarged or reduced.
ABC 
Triangle ABC has vertices A(0, 0), B(2, 6), and
ABC
C(6, 4). Find the coordinates of the vertices of
1
the image after a dilation with a scale factor .
2
Preimage
Image
ABC
ABC
1
1

A(0, 0)   0  , 0   
A(0, 0)
2
2

1
1

B(2, 6)   2  , 6    B(1, 3)
2
2

1
1

C(6, 4)   6  , 4    C(3, 2)
2
2

FEG  HEJ. Find the coordinates of F and
the scale factor.
FE EG
Write a proportion.

HE EJ
FE 4
HE  6, EG  4, and EJ  8.

6
8
8(FE)  24
Cross Products Property
FE  3
Divide both sides by 8.
So the coordinates of F are (0, 3). Since F(0, 3)  (0 • 2, 3 • 2)  H(0, 6), the
2
scale factor is .
1
Dilations and Similarity in the Coordinate Plane
You can prove that triangles in the coordinate plane are similar by using
the Distance Formula to find the side lengths. Then apply SSS Similarity
or SAS Similarity.
Use the figure to prove that ABC  ADE.
Step 1
Determine a plan for proving the
triangles similar.
AB
A  A by the Reflexive Property. If

AD
AC
, then the triangles are similar by SAS .
AE
Step 2
Use the Distance Formula to find
the side lengths.
AB 
1  3
2
  4  1
2
AC 
 13
AD 
Step 3
 1  3 
5  3
2
  3  1
2
 8 2 2
2
  7  1
2
AE 
7  3
2
  5  1
2
 52  2 13
 32  4 2
Compare the corresponding sides to determine whether they
are proportional.
AB
13
1


AD 2 13 2
AC 2 2 1


AE 4 2 2
The similarity ratio is
1
AB AC
, and
. So ABC  ADE by SAS .

2
AD AE
5. Prove that FGH  FLM.
________________________________________
________________________________________
6. Prove that QRS  TUV.
________________________________________
Name _______________________________________ Date __________________ Class __________________
Triangle Similarity: AA, SSS, and SAS
Angle-Angle (AA)
Similarity
Side-Side-Side (SSS)
Similarity
Side-Angle-Side (SAS)
Similarity
If two angles of one triangle
are congruent to two angles of
another triangle, then the
triangles are similar.
If the three sides of one
triangle are proportional to the
three corresponding sides of
another triangle, then the
triangles are similar.
If two sides of one triangle are
proportional to two sides of
another triangle and their
included angles are
congruent, then the triangles
are similar.
ABC  DEF
ABC  DEF
ABC  DEF
Explain how you know the triangles are similar, and write a similarity
statement.
1.
2.
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
3. Verify that ABC  MNP.
________________________________________
________________________________________
Name _______________________________________ Date __________________ Class __________________
Choose the best answer.
6. Complete the similarity statement.
1. Two points on a line are M(4, 5) and
N(2, 9). Which ratio represents the
slope of the line?
A 
B
2
3
1
7
C
2
3
D 7
2. The ratio of the angle measures of a
pentagon is 4 : 5 : 3 : 8 : 7. What is
the measure of the smallest angle?
F 27°
H 60°
G 40°
J 81°
3. Three sides of a triangle measure
0.5, 0.6, and 0.8. Two sides of a similar
triangle measure 2 and 3.2. What is
the length of the third side?
A 0.06
C 1.28
B 0.8
D 2.4
4. An Eiffel Tower mural is 2 meters high
and 0.77 meters wide. If the actual tower
is 125 meters wide, how tall is it to the
nearest meter?
F 48 m
H 325 m
G 125 m
J 842 m
ABC  ?
F ZYX
H XZY
G YZX
J XYZ
7. Rob is making for his young son a replica
armchair of one he uses. The ratio of the
length of the arm of the old chair to that of
the new chair is 3 : 2. The arm of the old
chair is 9 inches long. How many inches
long will the arm of the new chair be?
A 6
C 13
B 8
D 18
1
2
8. Which two triangles could you prove
similar by AA ?
F
H
G
J
5. Which is similar to this quadrilateral?
9. To measure the distance across a pond,
a surveyor locates points A, B, C, D, and
E as shown. What is AB to the nearest
meter?
A
C
B
D
A 10
C 15
B 12
D 18
Name _______________________________________ Date __________________ Class __________________
Answers for Multiple qustions
1. A
6. F
2. H
7. A
3. D
8. F
4. H
9. C
5. B