Download Section 5.1

5.1 – Congruence and similarity for 2D figures,
Properties of right triangles, Dilatations
Curriculum Outcomes:
D2
apply the properties of similar triangles
D6
solve problems involving measurement using bearings and vectors
D7
determine the accuracy and precision of a measurement (optional)
D8
solve problems involving similar triangles and right triangles
D9
determine whether differences in repeated measurements are significant or accidental
E8
use inductive and deductive reasoning when observing patterns, developing properties,
and making conjectures
Congruent and Similar Triangles
The term congruent means that two figures have the same shape and are the same size. The two
line segments are congruent if they have the same measure. The two congruent figures can fit
exactly on top of each other. Therefore, two triangles are congruent if the lengths of all sides and
the measure of all angles of one triangle are equal to all the angles and corresponding sides of the
second triangle. (Note: congruency applies to shapes other than triangles).
The following pairs of triangles represent congruent triangles.
D
A
F
ΔABC  ΔDEF
C
B
B
E
F
ΔABC  ΔDEF
A
D
C
E
Triangles can also be similar. Similar triangles have the same angles but their corresponding
sides are in the same ratio. The two triangles will be an enlargement or a reduction of one
another with all sides having the same scale factor. This enlargement or reduction will result
because of a dilatation. A dilatation is a transformation that stretches or shrinks a function, a
graph, or a figure, by the same scale factor. Here are some examples of similar triangles.
Triangle 1
Triangle 2
8
4
6
12
The corresponding sides of Triangle 1 and triangle 2 are in the same ratio.
4 6

8 12
1 1

2 2
The scale factor of the reduction is
1
.
2
Example1: The two triangles are similar and the scale factor of the enlargement is
Find x.
2
3
x
5
3
.
2
Solution 1:
2 x

3 5
2  x
5    5
3 5
10 5 x

3
5
10
x
3
1
3 x
3
x3
1
3
Similar triangles have the same shape but not the same size. The symbol for similar is . Two
triangles are similar if their corresponding angles are congruent (equal) and the ratio of their
corresponding sides is in proportion.
B
E
6
10
12
20
A
C
7
D
F
14
ΔABC  ΔDEF
 A = D
B = E
C = F
AB AC BC


DE DF EF
B
Sometimes similar triangles sit on top of each other.
To show that these triangles are similar, we need
only show that two angles of ΔBDE equal two
angles of ΔABC.
D
E
BDE = BAC because of corresponding angles
B is common to both triangles
C
A
Example 2: Solve for BE.
B
8
D
16
Solution 2:
x
E
18
8
x

16 18
8
x

24 x  18
8  x
18    18
 16   18 
8x  18  24 x
144
x
16
8x + 144 = 24x
x9
A
C
8x – 8x + 144 = 24x – 8x
144 = 16x
144 16 x

16
16
x9
Example 3: Solve for EC.
B
Solution 3:
24
12
D
12
24

30 24  x
12 24

18
x
 12   24 
x     x
 18   x 
E
12 x
 24
18
x
18
 12 x 
18
  24  18
 18 
C
A
1224  x  720
288 + 12x = 720
12x = 432
12 x  432
12 x 432

12
12
x  36
Example 4: Solve for DE.
B
Solution 4:
10
x

18 25
10
D
x
 10   x 
25    25
 18   25 
E
250
x
18
18
x  13.8
25
A
C
x = 36
Exercises:
1. Of right triangles, isosceles triangles, scalene triangles, and equilateral triangles, which
are always similar triangles? Explain.
2. The sides of a triangle are 7, 8, and 12. Find the length of the longest side of a similar
triangle whose shortest side is 21.
B
3.
What is the length of BC?
4
D
6
A
E
8
C
4. Two ladders lean against a wall as shown in the diagram. Each ladder makes the same angle
with the ground. The shorter ladder reaches 3.5m up the wall. How much further up the wall will
the other ladder reach?
5. The shadow of a boy 1.7m tall is 2.6m. At the same time, the shadow of a tree is
9.3m. How tall is the tree?
Answers:
1. Equilateral triangles are always similar triangles because all sides have the same length.
2.
E
B
8
7
21
A
12
C
D
F
x
7 1

Ratio of sides:
21 3
12 1

x 3
Longest Side:
x = 36
3.
B
4
D
6
A
4
x

10 x  8
E
8
C
BC = x + 8
4x + 32 = 10x
BC = 5.3 + 8
4x – 4x + 32 = 10x – 4x
BC = 13.3
32 6 x

6
6
x = 5.3
4.
3.5
4

3.5  x 9.5
14 + 4x = 33.25
14 – 14 + 4x = 33.25 – 14
4x = 19.25
4 x 19.25

4
4
x = 4.81
The longer ladder will extend 4.81m further up the wall.
5.
1 .7
x

2 .6 9 .3
 1.7   x 
9.3

9.3
 2.6   9.3 
15.81
x
2.6
x  6.08 m
The tree is 6.08 m tall.