Honors Geometry Section 5.4

The Pythagorean Theorem

In a right triangle the two sides that form the right angle are called

the legs, while the side opposite the right angle is called the

hypotenuse.

Consider placing four congruent right triangles with legs a and b and hypotenuse c as shown at the right. Notice that the large figure is a square. Using the formula for the area of a square (A = s2) what is its area?

))((

)( 2

babaA

baA

22 bababaA 22 2 babaA

We can also find the area of the large figure by adding the areas of the smaller square and the four triangles. The area of a triangle is found by the formula .

bhA 21

)21(44

2

abA

cA

triangles

aresmallersqu

abcA 22

If we set the two expressions for the area of the larger square equal to each other, we get:

abcbaba 22 222

222 cba

The Pythagorean Theorem For any right triangle with hypotenuse c and legs a and b, the sum of the squares of the legs (

)is equal to the square of the hypotenuse ( ).22 ba

2c

222 cba

54

516

80

80

842

222

x

x

x

x

x

576

576

2572

222

x

x

x

bhA2

1

847242

1A

24

A Pythagorean Triple is three whole numbers that could be the

sides of a right triangle.

15,12,9

,10,8,6

5,4,3

39,36,15

,26,24,10

13,12,5

75,72,21

50,48,14

25,24,7

Example: If a 25-foot ladder is leaning against a house and the bottom of the ladder is 9 feet away from the house, how far up the side of the house is the top of the ladder? Round to the nearest 1000th.

324.23544

544

2592

222

x

x

x

The converse of the Pythagorean Theorem is also true.

Pythagorean Theorem Converse

If the square of the largest side of a triangle equals the sum of the squares of the other two sides,

then the triangle is a right triangle.

ngle.right tria a is ABC then ,c If 222 ba

If a triangle is not a right triangle, then it must be either acute or

obtuse.

triangle.obtusean is ABC then ,c If 222 ba

triangle.acutean is ABC then ,c If 222 ba

Examples: Is a triangle with the given sides acute, right, obtuse or can’t exist. If the triangle cannot exist, explain why.

exist.cannot triangle then thesides, other two theof sumsidelongest If

53 64

27 8 222

obtuse

8.94 11.18 47.4

100 125

80 20 125

5452 55222

obtuse

Examples: Is a triangle with the given sides acute, right, obtuse or can’t exist. If the triangle cannot exist, explain why.

07.7

61 50

65 25 222

acuteexistnot Does

1164

73

73

382

222

AC

AC

AC

_____73 AC 11