About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.

Pythagorus realized that if you have a right triangle,

3

4

5

and you square the lengths of the two sides that make up the right angle,

24233

4

5

and add them together,

3

4

5

2423 22 43

22 43

you get the same number you would get by squaring the other side.

222 543 3

4

5

Is that correct?

222 543 ?

25169 ?

It is. And it is true for any right triangle.

8

6

10222 1086

1006436

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 which come together in a right angle are called

The lengths of the legs are usually called a and b.

a

b

The side across from the right angle

a

b

is called the

a2 + b2 = c2

Used to find a missing side of a right triangle

a & b always shortest sides * c is always longest side

Steps

1. Identify what sides you have and which side you are looking for.

2. Substitute the values you have into the appropriate places in the Pythagorean Theorem a2 + b2 = c2

3. Do your squaring first… then solve the 2-Step equation.

TOTD: if your answer under the radical is not a perfect square, leave your answer under the radical.

4

5

c

6.40 c

A.

Pythagorean TheoremSubstitute for a and b.

a2 + b2 = c2

42 + 52 = c2

16 + 25 = c2

41 = c

Simplify powers. Solve for c; c = c2.

Example 1A: Find the Length of a Hypotenuse

Find the length of the hypotenuse.

41 = c2

Example: 2 Finding the Length of a Leg in a Right Triangle

25

7

b

576 = 24b = 24

a2 + b2 = c2

72 + b2 = 252

49 + b2 = 625–49 –49

b2 = 576

Solve for the unknown side in the right triangle.

Pythagorean TheoremSubstitute for a and c. Simplify powers.

Try This: Example 1A

5

7

cA.

Find the length of the hypotenuse.

8.60 c

Pythagorean TheoremSubstitute for a and b.

a2 + b2 = c2

52 + 72 = c2

25 + 49 = c2

74 = cSimplify powers. Solve for c; c = c2.

Try This: Example 2

b 11.31

12

4

ba2 + b2 = c2

42 + b2 = 122

16 + b2 = 144–16 –16

b2 = 128

128 11.31

Solve for the unknown side in the right triangle.

Pythagorean TheoremSubstitute for a and c. Simplify powers.

15 = c

B.

Pythagorean TheoremSubstitute for a and b.

a2 + b2 = c2

92 + 122 = c2

81 + 141 = c2

225 = cSimplify powers. Solve for c; c = c2.

Example 1B: Find the the Length of a Hypotenuse

Find the length of the hypotenuse.

triangle with coordinates

(1, –2), (1, 7), and (13, –2)

B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2)

x

y

The points form a right triangle.

(–2, –2)

(–2, 4)

(3, –2)

Try This: Example 1B

Find the length of the hypotenuse.

7.81 c

Pythagorean Theorema2 + b2 = c2

62 + 52 = c2

36 + 25 = c2

61 = cSimplify powers. Solve for c; c = c2.

Substitute for a and b.

Example 3: Using the Pythagorean Theorem to Find Area

a6 6

4 4

a2 + b2 = c2

a2 + 42 = 62

a2 + 16 = 36

a2 = 20a = 20 units ≈ 4.47 units

Find the square root of both sides.

Substitute for b and c.Pythagorean Theorem

A = hb = (8)( 20) = 4 20 units2 17.89 units212

12

Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle.

a2 + b2 = c2

a2 + 22 = 52

a2 + 4 = 25

a2 = 21

a = 21 units ≈ 4.58 units

Find the square root of both sides.

Substitute for b and c.

Pythagorean Theorem

A = hb = (4)( 21) = 2 21 units2 4.58 units212

12

Try This: Example 3

Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle.

a5 5

2 2

Lesson Quiz

1. Find the height of the triangle.

2. Find the length of side c to the nearest meter.

3. Find the area of the largest triangle.

4. One leg of a right triangle is 48 units long, and the hypotenuse is 50 units long. How long is the other leg?

8m

12m

60m2

14 units

h

c10 m

6 m 9 m

Use the figure for Problems 1-3.