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The. Pythagorean. Theorem. c. a. b. This is a right triangle:. We call it a right triangle because it contains a right angle. The measure of a right angle is 90 o. 90 o. The little square. in the. angle is telling you that it is a. right angle. 90 o.
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The Pythagorean Theorem c a b
We call it a right triangle because it contains a rightangle.
The little square in the angle is telling you that it is a right angle. 90o
About 2,500 years ago, a Greek mathematician named Pythagoras discovered a special relationship that exists between the three sides of every right triangle.
5 3 4 Pythagorus realized that if you have a right triangle,
5 3 4 when you square the lengths of the two sides that make the right angle,
5 3 4 and then add the squares together,
5 3 4 the sum is the same value you get when you square the longest side.
Is that correct? ? Does: ? √
10 8 6 It is, and the same is true for any right triangle. √
The two sides which come together in a right angle are called
The two sides which come together in a right angle are called
The two sides that together form the right angle are called the LEGS.
The side across from the right angle is called the hypotenuse. a b
And the length of the hypotenuse is usually labeled c. c a b
The relationship Pythagoras discovered is now called The Pythagorean Theorem: c a b
The Pythagorean Theorem states that, given a right triangle with legs a and b and hypotenuse c, c a b
then . . . c a b
then . . . = c2 c a2 a b + b2
then . . . = 52 = 25 16 42 4 5 3 + 32 9
You can use The Pythagorean Theorem to solve many kinds of problems. Suppose you drive directly west for 48 miles, 48
Using The Pythagorean Theorem, 48 482 + 362 = c2 36 c
48 482 + 362 = c2 36 c Why? Can you see that we have a right triangle?
48 482 + 362 = c2 36 c Which side is the hypotenuse? Which sides are the legs?
Then all we need to do is calculate: √ √ 60 = c
So, since c2 is 3600, c is 60. And you end up 60 miles from where you started. So, since c2 is 3600, c is 48 36 60
15" 8" Find the length of a diagonal of the rectangle: ?
15" 8" Find the length of a diagonal of the rectangle: ? b = 8 c a = 15
c b = 8 a = 15
15" 8" Find the length of a diagonal of the rectangle: 17”
Practice using The Pythagorean Theorem to solve these right triangles:
c 5 12 = 13
b 10 26
b 10 26 Think: c2 b2 = a2 +
b 10 26 So: c2 - a2 b2 =
b 10 26 (a) (c)
b 10 26 (a) (c) 262 b2 = 102 +
b 10 26 (a) (c) 676 - b2 100 =
b 10 26 (a) (c) b2 576 = 24 √
b 10 26 = 24 (a) (c) c2 676 √ b2 576 = + a2 100
12 b 15 Your Turn! a2 + b2 = c2 (a) a = 12 c = 15 Awesome! = 9 (c) (b) a2 + b2 = c2 (12)2 + b2 = (15)2 (144) + b2 = (225) b2 = (225) - (144) b2 = 81 b = √81
Your Turn! a = 24 b = 32 Find the length of the diagonal. 32 in a2 + b2 = c2 (b) (24)2 + (32)2 = c2 24 in (a) =40 (c) (576) + (1024) = c2 c2 = (1600) √c2 = √1600 c = 40 The length of the diagonal is 40 inches.