the pythagorean theorem

THE PYTHAGOREA

N THEOREM

The Pythagorean Theorem

Named for Pythagoras, a Greek mathematician who lived in the sixth century B.C.

Babylonians, Egyptians, and Chinese were aware of this relationship

The Pythagorean Theorem

In a right triangle, the side opposite the right angle is the longest side. It is the hypotenuse. The other two sides are the legs of a right triangle.

legs

hypotenuse

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

a2 + b2 = c2

The Pythagorean Theorem

a

b

c

TRY THESE: A right triangle has sides of lengths 20,

29, and 21. What is the length of the hypotenuse?

Verify that the Pythagorean Theorem is true for the right triangle in the previous question.

Find the length of the hypotenuse of a right triangle with legs of lengths 7 and 24.

WHO? Czech-American mathematician Olga Taussky-Todd (1906-1995) studied Pythagorean triangles. In 1970, she won the Ford Prize for her research

SOMETIMES YOU WILL LEAVE YOUR ANSWER IN SIMPLEST RADICAL FORM.

Find the value of x. Leave your answer in simplest radical form.

The hypotenuse of a right triangle has length 12. one leg has length 6. Find the length of the other leg in simplest radical form.

820

x

Find the value of x.

1. 2.

3. 4.

x

8

6

x

x

x x8

x26 26

48

2x

3

3

When the lengths of the sides of a right triangle are integers, the integers form a Pythagorean Theorem. Here are some common primitive Pythagorean Triples.

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

Choose an integer. Multiply each number of a Pythagorean triple by that integer. Verify that the result is a Pythagorean triple.

9, 40, 41 11, 60, 61 12, 35, 37 13, 84, 85

What is the length of the diagonal of a rectangle whose sides measures 5 and 7?

Calculate the length of the side of a square whose diagonal measures 9 cm.

What is the measure of the longest stick we can put inside a 3 cm x 4 cm x 5 cm box?

Pythagorean Theorem & Other Shapes

Pythagorean Theorem ConverseIn ΔABC with longest side c, if c2 = a2 + b2, then the triangle is right. if c2 > a2 + b2, then the triangle is

obtuse. if c2 < a2 + b2, then the triangle is

acute. B

C A

a

b

c

a) 2, 3, 4

b) 3, 4, 5

c) 4, 5, 6

d) 3, 3, 3

e) 3, 3, 3

f) 2, 2, 4

g) 5, 5, 5

h) 4, 4, 5

i) , , 2

j) 2.5, 6, 6.5

The number represent the lengths of the sides of a triangle (a, b, c). Classify each triangle as acute, obtuse, or right.

obtuse

right

acute

right

obtuse

right

acute/equi

acute

right

right

a) 2, 3, 4

b) 3, 4, 5

c) 4, 5, 6

d) 3, 3, 3

e) 3, 3, 3

f) 2, 2, 4

g) 5, 5, 5

h) 4, 4, 5

i) , , 2

j) 2.5, 6, 6.5

The number represent the lengths of the sides of a triangle (a, b, c). Classify each triangle as acute, obtuse, or right.

obtuse

right

acute

right

obtuse