Algebra

12.5 The Pythagorean Theorem

Radical Review

Simplify each expression. 2

63

2 )2

2)5( )1

2

5

3 )4

You try!

2)7(2 )3

= 5 = 8/3

= 28 = 9/5

Do you know these?

7 =2

49

8 =2

64

14 =2

196

15 =2

225

12 =2

144 10 =2

100

13 =2

169

9 =2

81

11 =2

121 17 =2

289

16 =2

256

Pythagorean Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

a

bc

222 cba

What is the hypotenuse of a right triangle?

It is very important to identify the hypotenuse when using the Pythagorean Theorem.

Hypotenuse

1) The hypotenuse is always the longest side of the triangle.

2) The hypotenuse is always opposite the right angle.

Find the value of x together.

C A

B# BC AC AB

1) 8 6 x

2) x 9 15

3) 12 x

x = 10

x = 12

x = 38 34

Who can write the steps on the board?

Find the value of x on your own.

C A

B# BC AC AB

4) 5 5 x

5) x 3

Who can solve these on the board?

x =

x =

25

7) 1 x x =

6) x 1 x = 1 2

8) x 8 x =

6

12

2

33

3

54

A right triangle has one leg that is 3 inches longer than the other leg. The hypotenuse is 15 inches. Find the missing lengths.

x + (x + 3) = 152 2 2

x2 + x2 + 6x + 9 = 225

2x + 6x – 216 = 02

x

x + 3

152

x + 3x – 108 = 02

(x – 9)(x + 12) = 0

x = 9, -12

Lengths must be positive!

9

12

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Determining Right Triangles

bcIf c² = a² + b² Rt. ∆

a

If a ∆ is formed with sides having lengths of 4, 7 and 9, is it a right ∆ ?

Identify the longest side. This is the potential hypotenuse.

??? a2 + b2 = c2 ???.

??? 42 + 72 = 92 ???.

??? 16 + 49 = 81 ???.

??? 65 = 81 ???.

The triangle is not a right triangle..

Pythagorean Triples

3,4,5 5,12,13 8,15,17 7,24,25

6, 8, 10 9, 12, 1512, 16, 2015, 20, 25 etc.

10, 24, 2615, 36, 3920, 48, 5225, 60, 65 etc.

16, 30, 3424, 45, 5132, 60, 68 etc.

14, 48, 5021, 72, 7528, 96, 100 etc.

Memorize the 4 special triples at the top. Use them to save time and effort.

A Few from the HW

P. 741 #23 and P. 742 #29

HW

P. 741-742 #13-30, 45-46