Slide 1

11-3 The Pythagorean Theorem and the Distance Formula Special Right Triangles

Converse of the Pythagorean Theorem

The Distance Formula: An Application of the Pythagorean Theorem

Slide 2

Parts of a Right Triangle

Hypotenuse – side opposite of the right angle.

Legs – are the other two sides.

Slide 3

Pythagorean Theorem

Given a right triangle with legs a and b and hypotenuse c, then c2 = a2 + b2.

If BC = 3 cm and AC = 4 cm,

what is the length of AB?

Answer:

2 2 2

2 2 2

23 49 16 25

25 5

c a bccc AB

4 =

3

= 5

Slide 4

Proof of the Pythagorean Theorem

( )2

2 2

2 2

2 2 2

2 2 2

( )( )

2

2 2

a b

a b a b

a ab ab b

a ab b

c ab a ab b

c a b

+

+ +

+ + +

+ +

+ = + +

= +

Slide 5

Proof of the Pythagorean Theorem The square on one leg that is labeled 1 could be cut off and placed in the dashed space on the square of the hypotenuse. Then pieces 2, 3, 4, and 5 could be cut off and placed around piece 1 so that the square on the hypotenuse is filled exactly with five pieces. This shows that the sum of the areas of the squares on the two legs of a right triangle is equal to the area on the square of the hypotenuse.

Slide 6

Use the Pythagorean theorem to find the height of the trapezoid:

52 + h2 = 132 25 + h2 = 169 h2 = 169 – 25h2 = 144h = = 12

ExampleFind the area of the trapezoid.

Solution

h

A = ½ h (b1 + b2)

A = ½(12)(10 + 18)

A = 168 cm2

The length of the bottom base is 5 + 10 + 3 = 18 cm. So the area of the trapezoid is 168 cm2

144

Slide 7

Special Right Triangles45° -45° -90° right triangle

The length of the hypotenuse in a 45° -45° -90° right triangle is times the length of a leg.2

Slide 8

30° -60° -90° right triangle

In a 30°-60°-90° right triangle, the length of the hypotenuse is twice the length of the leg opposite the 30° (the shorter leg. The leg opposite the 60° angle (the longer leg) is times the length of the shorter leg.

3

Slide 9

Converse of the Pythagorean Theorem

If ∆ABC is a triangle with sides of lengths a, b, and c such that c2 = a2 + b2, then ∆ABC is a right triangle with the right angle opposite the side of length c.

Example

Determine if a triangle with the lengths of the sides given is a right triangle.

a. 20, 21, 29

b. 5, 18, 25

Answer: Yes, 202 + 212 = 841 = 292

Answer: No, 52 + 182 = 349 252

Slide 10

The Distance Formula: An Application of the Pythagorean Theorem

2 2 2

2 2 2

2 2

2

3 4

5

25

25

5

CD DE CE

CD

CD

CD

CD

CD

= +

= +

=

=

=

=

Slide 11

The Distance Formula

The distance between the points A(x1, y1) and B(x2, y2) is

( ) ( )

( ) ( )

2 22 1 2 1

2 222 1 2 1

AB x x y y

AB x x y y

= - + -

= - + -

Slide 12

Example

Determine what kind of triangle is formed by joining the points A(4, 7), B(–4, –7), and C(–7, 4).

Solution

So ∆ABC is an isosceles right triangle.

Slide 13

HOMEWORK 11-3

Pages 781 – 783

# 1, 3, 7, 9, 17