Chapter 6 Similarity of Figures
Similar Triangles
NEW SKILLS: WORKING WITH SIMILAR TRIANGLES
Similar triangles are very useful in making calculations and determining measurements.
The sum of the angles of a triangle is always 180°. If two corresponding angles in two
triangles are equal, the third angles will also be equal.
Two triangles are similar if any two of the three corresponding angles are congruent, or
one pair of corresponding angles is congruent and the corresponding sides adjacent to
the angles are proportional.
Two right triangles are similar if one pair of corresponding acute angles is congruent.
Example 1
Given the two triangles below, find the length of n.
C
b
a = 5 in
N
m
ℓ = 2 in
B
x
c = 7 in
A
M
x
n
SOL U T ION
You know that two of the three corresponding angles are congruent.
∠C = ∠N
∠B = ∠M
This means that ∆ABC is similar to ∆LMN.
L
6.4
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MathWorks 10 Workbook
To solve for n, set up a proportion.
n = c
a
n = 2
7 5
5× 7 × n = 2 ×7× 5
7
5
5n = 7 × 2
5n = 14
n = 14
5
n = 2.8
Side n is 2.8 in long.
For more information, see pages 257–258 of MathWorks 10.
BUILD YOUR SKILLS
1. In each of the diagrams below, ∆ABC is similar to ∆XYZ. Find the length of the
indicated side (to one decimal place).
A
a)
Z
x
6.1 cm
Y
4.5 cm
X
B
5.3 cm
C
b)
X
12.7 ft
A
x
B
Y
18.8 ft
8.2 ft
Z
C
Chapter 6 Similarity of Figures
c)
X
B
c
279
4m
C
A
25 m
Y
16 m
Z
2. Given that ∆ABC in similar to ∆RST, AB is 6 cm long, BC is 5 cm long, and RS is
8 cm long, find the length of a second side in ∆RST. Can you find the length of the
third side? Explin your answer.
3. C
armen thinks that any two isosceles triangles will be similar. Use examples to
prove or disprove her belief.
An isosceles triangle
has two sides equal
in length, and two
angles of equal
measure.
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MathWorks 10 Workbook
Example 2
Ravi notices that a 2-m pole casts a shadow of 5 m, and a second pole casts a shadow of
9.4 m. How tall is the second pole?
SOL U T ION
Sketch the situation.
sun
sun
pole 1 = 2 m
pole 2 = x
shadow = 5 m
shadow = 9.4 m
The angle between the rays of the sun and the pole is the same in both cases, so the
two triangles are similar.
Set up a proportion to solve for the height of the second pole.
height of pole 1 height of pole 2
=
shadow 2
shadow 1
2 = x
5 9.4
4
5 × 9.4 × 2 = x × 9.4 × 5
5
9.4
9.4 × 2 = 5 x
18.8 = 5 x
3.8 ≈ x
The second pole is approximately 3.8 m tall.
Chapter 6 Similarity of Figures
BUILD YOUR SKILLS
4. Assuming that the slope of a hill is constant, and that a point 100 metres along the
surface of the hill is 4.2 metres higher than the starting point, how high will you be
if you walk 250 metres along the slope of the hill?
5. Maryam is sewing a patchwork quilt. The sketch she has drawn is to a scale
of 1:8. Part of the design consists of right triangles that have legs that are
2.2 cm and 4.6 cm long. What will the lengths of the legs of the triangles in the
finished quilt be?
6. Which of the following triangles are similar?
C
45° 45°
A
D
B
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MathWorks 10 Workbook
PRACTISE YOUR NEW SKILLS
1. In the following diagram, AB is parallel to ED, AB is 8 m, AC is 12 m, and CE is
7 m. Calculate ED to one decimal place.
8m
B
A
12 m
C
7m
E
D
2. Madge has cut out two triangular shapes from a block of wood, as shown below. She
says that the two shapes are similar. Is she correct? Show your calculations.
16 in
8 in
2 in
10 in
8 in
2 in
5 in
3. Given that ∆FGH ~ ∆XYZ, state which angles are equal and which sides are
proportional.
Chapter 6 Similarity of Figures
4. Julian is visiting the Manitoba Legislative Building in Winnipeg, where he sees the
statue of Louis Riel. Use the information in the diagram to find the height of the
statue. Round your answer to a whole number.
x
Julian
6 ft
5 ft 8 in
2 ft
8 ft
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