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55 3344
9 square units
9 square units
25 square units
25 square units
16 square units
16 square units
55 3344
9 square units
9 square units
25 square units
25 square units
16 square units
16 square units
+ =+ =
The Pythagorean TheoremThe Pythagorean Theorem
If the hypotenuse of a right triangle has length c, and the
legs have lengths a and b, then a2 + b2 = c2.
If the hypotenuse of a right triangle has length c, and the
legs have lengths a and b, then a2 + b2 = c2.
Find the hypotenuse of a right triangle with legs of 8 and 15.
Find the hypotenuse of a right triangle with legs of 8 and 15.
c = 17c = 17
c2 = a2 + b2c2 = a2 + b2
c2 = 82 + 152c2 = 82 + 152
c2 = 64 + 225c2 = 64 + 225
c2 = 289c2 = 289√ c2 = √ 289√ c2 = √ 289
Example 1Example 1
Find the hypotenuse of a right triangle with legs of 6 and 7.
Find the hypotenuse of a right triangle with legs of 6 and 7.c2 = a2 + b2c2 = a2 + b2
c2 = 62 + 72c2 = 62 + 72
c2 = 36 + 49c2 = 36 + 49
c2 = 85c2 = 85√ c2 = √ 85√ c2 = √ 85
c = √ 85 ≈ 9.2c = √ 85 ≈ 9.2
Example 2Example 2
Find the hypotenuse of a right triangle with legs of 9 and 12.
Find the hypotenuse of a right triangle with legs of 9 and 12.
1515
ExampleExample
Find the hypotenuse of a right triangle with legs of
and .
Find the hypotenuse of a right triangle with legs of
and .
11
1212
√ 32
√ 32
ExampleExample
Find the hypotenuse of a right triangle with legs of 1 and 1.
Find the hypotenuse of a right triangle with legs of 1 and 1.
√ 2√ 2
ExampleExample
Find the leg of a right triangle whose hypotenuse is 16 and other leg is 7.
Find the leg of a right triangle whose hypotenuse is 16 and other leg is 7.a2 + 72 = 162a2 + 72 = 162 a2 + 49 = 256a2 + 49 = 256
a2 = 207a2 = 207a = √ 207 ≈ 14.4a = √ 207 ≈ 14.4
a2 + 49 – 49 = 256 – 49a2 + 49 – 49 = 256 – 49
Example 3Example 3
Find the length of a leg of a right triangle whose hypotenuse is 39 and whose other leg is 15.
Find the length of a leg of a right triangle whose hypotenuse is 39 and whose other leg is 15.
3636
ExampleExample
Find the length of a leg of a right triangle whose hypotenuse is 20 and whose other leg is 10.
Find the length of a leg of a right triangle whose hypotenuse is 20 and whose other leg is 10.
√ 300 ≈ 17.3√ 300 ≈ 17.3
ExampleExample
The converse is the statement resulting when the “if” part and the “then” part of a conditional statement are switched.
The converse is the statement resulting when the “if” part and the “then” part of a conditional statement are switched.
ConverseConverse
Converse of the Pythagorean Theorem
Converse of the Pythagorean Theorem
If a triangle has sides a, b, and c, such that
a2 + b2 = c2, then the triangle is a right triangle.
If a triangle has sides a, b, and c, such that
a2 + b2 = c2, then the triangle is a right triangle.
Determine whether a triangle with sides of 12, 35, and 37 is a right triangle.
Determine whether a triangle with sides of 12, 35, and 37 is a right triangle.
1,369 = 1,3691,369 = 1,369
a2 + b2 = c2a2 + b2 = c2
122 + 352 = 372122 + 352 = 372
144 + 1,225 = 1,369144 + 1,225 = 1,369
Example 4Example 4
yesyes
Determine whether a triangle with sides of 8, 12, and 14 is a right triangle.
Determine whether a triangle with sides of 8, 12, and 14 is a right triangle.
208 ≠ 196208 ≠ 196
a2 + b2 = c2a2 + b2 = c2
82 + 122 = 14282 + 122 = 142
64 + 144 = 19664 + 144 = 196
Example 5Example 5
nono
Determine whether a triangle with sides of 15, 18, and 22 is a right triangle.
Determine whether a triangle with sides of 15, 18, and 22 is a right triangle.
no; 152 + 182 ≠ 222no; 152 + 182 ≠ 222
ExampleExample
Determine whether a triangle with sides of 16, 30, and 34 is a right triangle.
Determine whether a triangle with sides of 16, 30, and 34 is a right triangle.
yes; 162 + 302 = 1,156 = 342yes; 162 + 302 = 1,156 = 342
ExampleExample
A 16 ft. ladder leans up against the side of a building. If the base of the ladder is 4 ft. from the base of the building, how high up the side of the building will the ladder reach?
A 16 ft. ladder leans up against the side of a building. If the base of the ladder is 4 ft. from the base of the building, how high up the side of the building will the ladder reach?
15.5 ft.15.5 ft.
ExerciseExercise
A 200 ft. tower is braced to the ground by a cable, from a point 150 ft. above the ground to a point 87 ft. from the base of the tower. How long is the cable?
A 200 ft. tower is braced to the ground by a cable, from a point 150 ft. above the ground to a point 87 ft. from the base of the tower. How long is the cable?
173.4 ft.173.4 ft.
ExerciseExercise
The distance between bases on a baseball diamond is 90 ft. How far is it from home plate to second base?
The distance between bases on a baseball diamond is 90 ft. How far is it from home plate to second base?
127.3 ft.127.3 ft.
ExerciseExercise
An opening for a window is 23” wide, 54” tall, and 60” diagonally. Is the opening “square”; that is, do the height and width form a right angle?
An opening for a window is 23” wide, 54” tall, and 60” diagonally. Is the opening “square”; that is, do the height and width form a right angle?
nono
ExerciseExercise
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