Geometric Mean & Geometric Mean & the Pythagorean Thm.the Pythagorean Thm.

Section 7-1 & 7-2

The geometric mean between 2 positive numbers a and b is the positive number x where

x2 = ab

x =

b

x

x

a

ab

Thm. 7-1Thm. 7-1 If the altitude is drawn from the

vertex of the right angle of a right triangle to its hypotenuse, then the 2 triangles formed are similar to the given triangle and to each other.

A BD

C

CDBADC ~

Thm. 7-2Thm. 7-2

The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the 2 segments of the hypotenuse.

a

h

b

b

h

h

a

Thm. 7-3Thm. 7-3

If the altitude is drawn to the hypotenuse of a right triangle, then the measure of a leg of the triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

d

h

e

a b

c

d

a

a

c

e

b

b

c

Pythagorean TheoremPythagorean Theorem

In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.

a

b

ca2 + b2 = c2

Converse is also true.

Pythagorean TriplesPythagorean Triples If the measures of a right triangle

are whole numbers, the measures of the sides are a Pythagorean triple.

3-4-5 5-12-13 8-15-17

7-24-25

Joke TimeJoke Time

What is a twip?

A twip is what a wabbit takes when he wides on a twain.

What is gray, has big ears and has a trunk?

A mouse going on vacation