1.3 Similar Triangles

1.3

Similar Triangles

The Great Hall of the National Gallery of Canada in Ottawa has an

impressive glass ceiling. The design of the Great Hall is intended to

mimic the design of the nearby Library of Parliament.

Architects use similar figures in scale drawings and scale models as

they design buildings.

Investigate

Tools

grid paper

protractor

ruler

Properties of Similar Triangles

Method 1: Use Pencil and Paper

1. On grid paper, draw triangles ABC, DEF, and GHI.

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2. Copy and complete the table.

¡ÏA =

¡ÏD =

¡ÏG =

¡ÏB =

¡ÏE =

¡ÏH =

¡ÏC =

¡ÏF =

¡ÏI =

AB =

DE =

GH =

BC =

EF =

HI =

AC =

DF =

GI =

1.3 Similar Triangles ? MHR

19

3. Which measures are equal?

Which are different?

4. Find the ratios of lengths of pairs of corresponding sides.

How do the ratios compare to one another?

How do the measures of corresponding angles compare?

Tools

Method 2: Use The Geometer¡¯s Sketchpad

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computers

The Geometer¡¯s

Sketchpad

?

1. Open The Geometer¡¯s Sketchpad . Open a new sketch.

?

2. From the Edit menu, choose Preferences. Click the Text tab. Make

sure that the Show labels automatically from all new points

check box is selected.

3. Use the Segment tool to construct a small triangle in the centre of

your screen.

4. Measure angle ABC: use the Selection tool to select points A, B,

C (in that order). Then, from the Measure menu, choose Angle.

Measure angles CAB and BCA.

5. Use the Text tool to label

sides a, b, and c of the

triangle ABC. Side a is

opposite angle A, side b

is opposite angle B and

side c is opposite angle C.

Using the Selection tool, select each line segment. Then, from the

Measure menu, choose Length. Record the side lengths.

20 MHR ? Chapter 1

6. Use the Point tool to construct a point inside your triangle.

Select this point. From the Transform menu, choose Mark

Center. Select every point and line segment on your triangle.

From the Transform menu, choose Dilate. Use a fixed ratio of

2 to 1. Click Dilate. You should have a new larger triangle on

your sketch. This new triangle is similar to triangle ABC.

Use the Text tool to label the sides of the new triangle.

corresponding angles

? have the same relative

position in a pair of

similar triangles

? are equal

ratio

? a comparison of two

quantities measured

in the same units

corresponding sides

? have the same relative

position in a pair of

similar triangles

7. Using the same method you used for the smaller triangle, measure

each of the angles and side lengths of the new, larger triangle.

What do you notice about the measures of corresponding angles ?

How do you think the ratios of the lengths of corresponding

sides compare?

8. To check your prediction about the ratios of lengths of

corresponding sides, select one of the sides of the larger triangle

and the corresponding side of the smaller triangle. From the

Measure menu, choose Ratio.

Repeat for the other pairs of corresponding sides.

How are these ratios related to each other?

Do you think this is true for all pairs of similar triangles?

9. Repeat step 6 using a different fixed ratio to construct another pair

similar triangles

? triangles in which the

ratios of the lengths

of corresponding

sides are equal and

corresponding angles

are equal

proportional

? two quantities are

proportional if

they have the same

constant ratio

? the side lengths of

two triangles are

proportional if there

is a single value which

will multiply all of the

side lengths of the

?rst triangle to get

the side lengths of

the second triangle

of similar triangles.

Measure the angles and sides of the new triangles and find the

ratios of the lengths of corresponding sides.

Compare your results with your classmates¡¯ constructions.

What patterns do you see?

Make a conclusion about the measures of the corresponding

angles of similar triangles.

Make a conclusion about the ratios of lengths of corresponding

sides of similar triangles.

Similar triangles

Triangles are similar if

? corresponding angles are equal

? corresponding sides are proportional in length

When naming similar triangles, list the letters of the corresponding

angles in the same order for both triangles. For example, given similar

triangles ABC and MNP, ¡ÏA corresponds to ¡ÏM, ¡ÏB corresponds to

¡ÏN, and ¡ÏC corresponds to ¡ÏP.

1.3 Similar Triangles ? MHR

21

Example

1

Find Missing Side and Angle Measures

Given
ABC ¡«
DEF, find the measure of ¡ÏC and the length of DE

to the nearest tenth of a unit.

Math Connect

#

The symbol

¡« means ¡°similar

to¡±. The symbol


means

¡°approximately

equal to¡±.



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Solution

Since
ABC ¡«
DEF, corresponding angles are equal.

Therefore, ¡ÏC = ¡ÏF

= 31¡ã

Corresponding sides are proportional in length.

DF

DE = _

Therefore, _

AB AC

16.1

DE = _

_

12.1 20.1

The numerators are the side

lengths of one triangle; the

denominators are the side

lengths of the other triangle.

12.1 ¡Á 16.1

DE = _

20.1


9.7

Multiply both sides by 12.1.

The measure of ¡ÏC is 31¡ã and the length of DE is 9.7 units.

Example

2

Use Opposite Angles and Similar Triangles

to Find Missing Measures

Find the length of FG to the nearest tenth of a unit.

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22 MHR ? Chapter 1

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Solution

If
FGJ ¡«
IHJ, then the lengths of corresponding sides are

proportional.

¡ÏHJI = 43¡ã (opposite angles)

¡ÏGFJ = 180¡ã - (43¡ã + 90¡ã)

= 47¡ã

¡ÏHIJ = 180¡ã - (43¡ã + 90¡ã)

= 47¡ã

Since ¡ÏGFJ = ¡ÏHIJ, ¡ÏFGJ = ¡ÏIHJ, and ¡ÏFJG = ¡ÏHJI, then


FGJ ¡«
IHJ.

FJ

FG _

_

=

HI

IJ

18.9

FG _

_

=

3.8

6.3

3.8 ¡Á 18.9

FG = _

6.3


11.4

Multiply both sides by 3.8.

Use a calculator.

The length of FG is 11.4 units.

Example

3

Use Parallel Lines and Similar Triangles

to Find Missing Measures

In the diagram, DE is parallel to AC.

Find the length of AC.

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DN

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DN

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Solution

Draw the triangles separately.

AD = 12 cm and

DB = 48 cm. AB is the

sum of the two lengths.

So, AB = 60 cm.

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DN


DN

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DN

$

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DN &

1.3 Similar Triangles ? MHR

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