Download Similar Triangles and the Pythagorean Theorem

Similar Triangles and the Pythagorean Theorem
Similar Triangles
Two triangles are similar if they contain angles of the same measure. Similar triangles
have the same shape but may be different in size. Also, the ratios of corresponding side
lengths of the triangles are equal.
SIMILAR TRIANGLE FACTS
If two triangles have three angles of the same measure, the triangles
are similar.
Similar triangles always have the same shape
If two triangles are similar, the sides are in direct proportion.
Example:
Example 1
In the above triangles,
6 in
3 in
10 in
5 in
8 in
4 in
2.
One triangle has angles of 45 , 45 , 90 and another triangle has angles of
40 , 50 , and 90 . Are they similar?
Since these triangles have different angle measures, they are not similar.
Example 2
Are the triangles shown here similar?
Since these triangles clearly have a different shape, they are not similar.
Example 3
Are the triangles shown here similar?
The ratio of the longest side lengths is 12 cm = 2.
The ratio of the shortest side lengths is
6 cm
6 cm
3 cm
= 2.
The ratio of the second longest side lengths is 9 cm = 2.
4.5 cm
Since the ratios are all the same, the triangles are similar.
Solving for Side Lengths of Similar Triangles
Many applications involve solving for side lengths of similar triangles. Since the ratio of
corresponding side lengths is constant in similar triangles, a proportion may be set up and
solved and an unknown side length may be determined.
PROCEDURE TO SOLVE FOR A SIDE LENGTH OF A SIMILAR TRIANGLE
1.
Write the ratios of corresponding side lengths. Use (?) or some
other variable for the unknown side length.
2.
Set two ratios equal to each other to make a proportion. Solve the
proportion for the unknown as was done in Section 5.1 .
Example 4
For the given similar triangles, find the unknown side length.
2 ft
2.1 ft
(?)
5 ft
The ratios of the side lengths are 2.1 ft and (?) ft .
5 ft
2 ft
Set these two ratios equal to each other to obtain
2.1 ft
5 ft
= (?) ft .
2 ft
Solve this proportion for the unknown by setting the cross products equal
and dividing.
2.1 × 2 = 5 × (?)
4.2 = 5 × (?)
4.2 ÷ 5 = (?) = 0.84 ft
Example 5
For the given similar triangles, find the unknown side length.
3.5 ft
1.4 ft
5.2 ft
The proportion consists of 3.5 ft = 5.2 ft .
1.4 ft
(?) ft
The unknown is determined by cross-multiplying and dividing.
1.4 × 5.2 = 3.5 × (?)
7.28 = 3.5 × (?)
7.28 ÷ 3.5 = (?) = 2.08 ft
(?)
Example 6
A building casts a 32 ft shadow. A 4 foot tall boy standing near the
building casts a shadow 6 ft in length. Using similar triangles, determine
the height of the building.
The two triangles illustrated are similar.
The proportion consists of 32 ft = (?) ft .
6 ft
4 ft
Solve this proportion for the unknown by making the cross products equal
and then dividing.
32 × 4 = 6 × (?)
128 = 6 × (?)
128 ÷ 6 = (?) = 21.33 ft (rounded to a hundredth of a foot)
Example 7
For the given similar triangles, find the unknown side length.
The proportion consists of 7.5 ft = 8.2 ft .
4.4 ft
(?) ft
4.4 × 8.2 = 7.5 × (?)
36.08 = 7.5 × (?)
36.08 ÷ 7.5 = (?) = 4.81 ft (rounded to a hundredth of a foot)
Right Triangles and the Pythagorean Theorem
A triangle is a right triangle if it contains a 90 degree angle. The Pythagorean
Theorem states the following relationship between the side lengths.
THE PYTHAGOREAN THEOREM
For any given right triangle with side lengths A, B, and C, where C
is the longest side, the following is always true.
C2
A2
C
B2
A
90
B
The lengths of any of the sides may be determined by using the
following formulas. Note that these formulas involve use of the
square root ( ) function on a calculator.
C
A2
B2
A
C2
B2
B
C2
A2
The longest side C is known as the hypotenuse of the triangle.
Example 8
Given the following right triangle, find the missing side length.
(?) in
5 in
12 in
Substituting A = 5 and B = 12 into the formula C
C
=
52
169
122
25
A2
B2
results in
144
= 13
The square root may be determined by entering the keystrokes
169
on a calculator.
Example 8
Given the following right triangle, find the missing side length. Round the
result to the nearest tenth of an inch.
9 in
5 in
(?) in
Substituting A = 5 and C = 9 into the formula B
B
92
52
=
56 = 7.48
81
C2
A 2 results in
25
7.5 inches .
The square root may be determined by entering the keystrokes
56
on a calculator.
MATH HINT
When calculating the side length of a right triangle, remember the
following.
(SHORT SIDE)2
(OTHER SHORT SIDE)2
(LONGEST SIDE)2
(OTHER SHORT SIDE)2
LONGEST SIDE
SHORT SIDE
Remember to square terms and add or subtract them before taking
the square root.
Example 9
The cross-section of a roof is shown below. What is the distance from the
peak of the roof to the edge of the roof?
(?) ft
(?) ft
8 ft
20 ft
20 ft
40 ft
This cross-section consists of two right triangles, each with short sides of
length 8 feet and 20 feet.
The longest side = (?) =
82
202
464
21.54 feet.