6.7Geometric Mean and the Pythagorean Theorem

Objectives:• To find the geometric mean between two numbers.• To solve problems involving relationships between

parts of a triangle and the altitude and its hypotenuse.

• To use the Pythagorean Theorem and its converse.

Vocabulary

• Geometric Mean• Pythagorean Triple

Geometric Mean

• Find the geometric mean between 2 and 10.– Let x represent the geometric mean.

47.45220

20

10

2

2

x

x

x

x

Example 1

• Find the geometric mean between 12 and 20.

Example 2

• Find the geometric mean between 6 and 15.

Right Triangle AltitudeSimilar Triangles Theorem

• If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the two right triangles formed are similar to the given triangle and to each other.

Right TriangleAltitude Theorem 1

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

Example 3

Right TriangleAltitude Theorem 2

• 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 segment of the hypotenuse adjacent to that leg.

Example 4• Find a and b in ∆TGR.

Pythagorean Triple• A Pythagorean Triple is a group of three whole

numbers that satisfies the equation

a2 + b2 = c2, where c is the greatest measure.

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

6-8-10 10-24-26 16-30-34

9-12-15

12-16-20

Homework

6.7 RSG