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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(
"
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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
?
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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o
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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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o
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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
&
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.
$
#
#
DN DN
"
DN
$
%
DN &
1.3 Similar Triangles ? MHR
23
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