Pythagorean Theorem and it’s Converse

Pythagorean TheoremPythagorean Theorem: used for right triangles, it is a tool used to

solve for a missing side of a right triangle

a2 + b2 = c2

cb

a

a, b are the legs of the trianglec must by the hypotenuse (the longest side of the triangle)

Examples…Find the missing side of the triangle.

1. 2.

The side missing is c because it is the side across from the right angle.

a2 + b2 = c2

92 + 122 = c2

81 + 144 =c2

225 = c2

225 = c2

15 = c

10 is c because it is across from the right angle. That means you are missing one of the legs (a).

a2 + b2 = c2

a2 + 82 = 102

a2 + 64 =100 - 64 - 64a2 = 36

a2 = 36a = 6

You try…Find the length of the missing side.

1. 2.

3. 4.

14

6

Answers and Solutions…1. 2.

3. 4.

a2 + b2 = c2

a2 + 122 = 132

a2 + 144 =169 - 144 -144a2 = 25

a2 = 25a = 5

a2 + b2 = c2

8.62 + 6.32 = c2

73.96 + 39.69 = c2

113.65 = c2

113.65 = c2

10.7 = c

a2 + b2 = c2

62 + b2 = 142

36 + b2 = 196-36 -36b2 = 160

b2 = 160b = 4 2 x 5b = 4 10 4 42 5

1610

160

a2 + b2 = c2

72 + 102 = c2

49 + 100 = c2

149 = c2

149= c2

149 = c

Application of Pythagorean Theorem

1. The foot of a ladder is placed 12 feet from a wall. If the top of the ladder rests 16 feet up on the wall, how long is the ladder?

2. David’s TV screen is 10 inches long. If the diagonal measures 25 inches, how long is the width of David's TV?

?16 ft

12 ft

wall ladder

a2 + b2 = c2

122 + 162 = c2

144 + 256 = c2

400 = c2

400 = c2

20 = cladder = 20 ft

? 25 in

10 in

a2 + b2 = c2

102 + b2 = 252

100 + b2 = 625-100 -100b2 = 525

b2 = 525b = 22.9width = 22.9 in

Converse of Pythagorean TheoremUsed to determine if 3 measurements create a right triangle.

If 3 measurements form a right triangle, then…

*because c must be the hypotenuse, the longest side given will always be used as c

a2 + b2 = c2

Determine if the following triangles are right triangles.1. 2.

a2 + b2 = c2

52 + 122 = 132

25 + 144 = 169169 = 169

True statement, so YES a right triangle.

a2 + b2 = c2

92 + 122 = 192

81 + 144 = 361225 = 361

False statement, so NO, not a right triangle.

You try…Determine whether the following triangles are right triangles.

1. 2.

Answers and Solutions…1. 2.a2 + b2 = c2

3.62 + 4.52 = 62

12.96 + 20.25 = 3633.21 = 36No, not a right triangle.

a2 + b2 = c2

5.62 + 92 = 10.62

31.36 + 81 = 112.36112.36 = 112.36Yes, a right triangle.

Pythagorean Triple

A set of positive integers that form a right triangle…

Most well known triples: 3, 4, 5

5, 12, 13

7, 24, 25

8, 15, 17

9, 40, 41

What type of triangle is it?Pythagorean Theorem can also be used to determine what type of triangle 3 measurements make.

a2 + b2 = c2 - right trianglea2 + b2 > c2 - acute trianglea2 + b2 < c2 - obtuse triangle

What type of triangle is it?

a2 + b2 = c2

92 + 122 = 142

81 + 144 = 196225 = 196225 > 196This is an acute triangle.

Determine what type of triangle is given.1. 2. 3.

a2 + b2 = 152

92 + 122 = 152

81 + 144 = 225225 = 225This is a right triangle.

a2 + b2 = 152

42 + 122 = 152

16 + 144 = 225160 = 225160 < 225This is an obtuse triangle.

You try…Determine whether each triangle is acute, right, or obtuse.

1. 2.

3. 4.

Answers and Solutions…1. 2.

3. 4.

a2 + b2 = 152

8.12 + 11.92 = 13.52

65.61 + 141.61 = 182.25207.22 = 182.25207.22 > 182.25This is an acute triangle.

a2 + b2 = 152

52 + 122 = 132

25 + 144 = 169169 = 169This is a right triangle.

a2 + b2 = 152

52 + 62 = 102

25 + 36 = 10061 = 10061 < 100This is an obtuse triangle.

a2 + b2 = 152

122 + 20.82 = 21.82

144 + 432.64 = 475.24576.64 = 475.24576.64 > 475.24This is an acute triangle.