12
Geometry
Copyright © Cengage Learning. All rights reserved.
12.3
Triangles
Copyright © Cengage Learning. All rights reserved.
Triangles
Triangles are often classified in two ways:
1. by the number of equal sides
2. by the measures of the angles of the triangle
Triangles may be classified or named by the relative
lengths of their sides.
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Triangles
In each triangle in Figure 12.28, the lengths of the sides are
represented by a, b, and c.
An equilateral triangle has all three sides
equal. All three angles are also equal.
An isosceles triangle has two sides equal.
The angles opposite these two sides are
also equal.
A scalene triangle has no sides
equal. No angles are equal either.
(a) Equilateral triangle
(b) Isosceles triangle
(c) Scalene triangle
Triangles named by sides
Figure 12.28
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Triangles
Triangles may also be classified or named in terms of the
measures of their angles (see Figure 12.29).
A right triangle has one right angle.
An acute triangle has three acute angles.
An obtuse triangle has one obtuse angle.
(a) Right triangle
(b) Acute triangle
(c) Obtuse triangle
Triangles named by angles
Figure 12.29
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Pythagorean Theorem
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Pythagorean Theorem
In a right triangle, the side opposite the right angle is called
the hypotenuse, which we label c.
The other two sides, the sides opposite
the acute angles, are called legs, which
we label a and b. (See Figure 12.29a.)
Right triangle
Figure 12.29(a)
The Pythagorean theorem relates the lengths of the sides
of any right triangle as follows:
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Pythagorean Theorem
The Pythagorean theorem states that the square of the
hypotenuse of a right triangle is equal to the sum of the
squares of the lengths of the two legs.
Alternative forms of the Pythagorean theorem are
a2 = c2 – b2
or
b2 = c2 – a2
or
and
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Example 1
Find the length of the hypotenuse of the triangle in
Figure 12.30.
Figure 12.30
Substitute 5.00 cm for a and 12.0 cm for b in the formula:
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Example 1
cont’d
Note:
You may need to use parentheses with some calculators.
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Perimeter and Area
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Perimeter and Area
To find the perimeter of a triangle, find the sum of the
lengths of the three sides.
The formula is P = a + b + c, where P is the perimeter and
a, b, and c are the lengths of the sides.
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Perimeter and Area
An altitude of a triangle is a line segment drawn
perpendicular from one vertex to the opposite side.
Sometimes this opposite side must be extended. See
Figure 12.34.
Figure 12.34
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Perimeter and Area
Look closely at a parallelogram (Figure 12.35a) to find the
formula for the area of a triangle.
Remember that the area of a
parallelogram with sides of
lengths a and b is given by A = bh.
Figure 12.35(a)
In this formula, b is the length of the base of the
parallelogram, and h is the length of the altitude drawn to
that base.
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Perimeter and Area
Next, draw a line segment from B to D in the parallelogram
as in Figure 12.35(b).
Two triangles are formed. We know from geometry that
these two triangles have equal areas.
Since the area of the parallelogram is
bh square units, the area of one
triangle is one-half the area of the
parallelogram.
Figure 12.35(b)
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Perimeter and Area
So the formula for the area of a triangle is
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Example 5
The length of the base of a triangle is 10.0 cm. The length
of the altitude to that base is 6.00 cm. Find the area of the
triangle. (See Figure 12.36.)
A=
bh
A=
(10.0 cm)(6.00 cm)
Figure 12.36
= 30.0 cm2
The area is 30.0 cm2.
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Perimeter and Area
If only the lengths of the three sides are known, the area of
a triangle is found by the following formula (called Heron’s
formula):
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Example 7
Find the perimeter and the area (rounded to three
significant digits) of the triangle in Figure 12.38.
P=a+b+c
P = 9 cm + 15 cm + 18 cm
Figure 12.38
= 42 cm
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Example 7
cont’d
To find the area, first find s.
s=
(a + b + c)
s=
(9 + 15 + 18)
=
(42)
= 21
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Example 7
cont’d
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Example 7
cont’d
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Perimeter and Area
The following relationship is often used in geometry and
trigonometry.
Figure 12.39
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