Transcript Using the Pythagorean Theorem
LESSON 3-7 Warm Up
Lesson 3-7 Warm Up
ALGEBRA READINESS
LESSON 3-7 Warm Up
Lesson 3-7 Warm Up
ALGEBRA READINESS
“Using the Pythagorean Theorem” (3-7)
What is the “Converse (Opposite) of the Pythagorean Theorem”?
Converse of the Pythorean Theorem:
If a triangle has sides a, b, and c, and a 2 + b 2 = c 2 (
in other words, if the sum of the squares of the shorter sides equal the square of the longest side
), then the triangle is a right triangle with hypotenuse c. The Converse (Opposite) of the Pythagorean Theorem can be use to determine whether or not a triangle is a right triangle.
Example: Determine whether the given lengths can form a right triangle.
5 in., 12 in., and 13 in.
a 2 + b 2 = c 2 Make a and b the two shorter sides .
7 in., 9 in., and 12 in.
a 2 + b 2 = c 2 Use the Converse of the Pythagorean Theorem
ALGEBRA READINESS
Using the Pythagorean Theorem LESSON 3-7 Additional Examples
Determine whether the given lengths can be sides of a right triangle.
a.
5 in., 5 in., and 7 in.
5 2 + 5 2 7 2 25 + 25 49 Determine whether
a
2 where
c
+
b
2 is the longest side.
=
c
2 , Simplify.
This triangle is not a right triangle.
b.
10 cm, 24 cm, and 26 cm
10 2 + 24 2 26 2 Determine whether
a
2 +
b
2 =
c
2 , where
c
is the longest side.
100 + 576 676 676 = 676 Simplify.
This triangle is a right triangle.
ALGEBRA READINESS
Using the Pythagorean Theorem LESSON 3-7 Additional Examples Is a triangle with sides 9 cm, 40 cm, and 41 cm a right triangle? Explain.
a
9 2 2 +
b
2 + 40 2 =
c
2 41 2 81 + 1,600 1,681 1,681 = 1,681 Use the Pythagorean Theorem.
The longest side, 41 cm, is the hypotenuse. Substitute 9 for
a
, 40 for
b
, and 41 for
c
.
Square 9, 40, and 41.
Simplify.
The equation is true, so the triangle is a right triangle.
ALGEBRA READINESS
Using the Pythagorean Theorem LESSON 3-7 Additional Examples The bottom of a 10-ft ladder is 2.5 ft from the side of a wall. How far, to the nearest tenth of a foot, is the top of the ladder from the ground?
The diagram shows a right triangle with hypotenuse 10 ft and leg 2.5 ft. The distance from the top of the ladder to the ground is
a
.
ALGEBRA READINESS
Using the Pythagorean Theorem LESSON 3-7 Additional Examples (continued)
a
2 +
b
2 =
c
2
a
2 + ( 2.5
) 2 = 10 2
a
2 + 6.25 = 100
a
2 = 93.75
a
2 = 93.75
a
= 93.75
Use the Pythagorean Theorem.
Substitute 2.5 for
b
and 10 for
c
.
Square 2.5 and 10.
Subtract 6.25 from each side.
Find the square root of each side.
Simplify.
This is about 9.7 feet.
ALGEBRA READINESS
Using the Pythagorean Theorem LESSON 3-7 Lesson Quiz 1.
A triangle has a hypotenuse of 17 in. and one of its legs is 8 in. What is the length of the other leg?
15 in.
2.
The bottom of a 12-ft ladder is 4 ft from the side of a house. Find the height of the top of the ladder above the ground.
√128 or 11.3 ft 3.
An artist is measuring a rectangular canvas. Its length is 30 in. The distance from one corner of the canvas to the other (along the diagonal) is 34 in. What is its width?
16 in.
4.
A triangular window has sides of lengths 12 in., 16 in., and 20 in. Is the window a right triangle? Explain.
Yes; the Pythagorean Theorem is true for 12 2 window is a right triangle.
+ 16 2 = 20 2 , so the ALGEBRA READINESS