GeometryLesson 8 – 1

Geometric Mean

Objective:Find the geometric mean between two numbers.

Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse.

Geometric Mean

Geometric meanThe positive square root of their product.

abxandabxsob

x

x

a 2,

Geometric mean between 9 and 4.

3636,4

9 2 xandxsox

x6

Find the geometric mean between 8 & 10.

abx

108x Simplify the radical!

5242 x

54x

What is the square root of 16?

Find the geometric mean

5 & 45 12 & 15

abx 455x

955 xSquare root of 25?Square root of 9?

x = 5(3)

x = 15

abx 1512 x

5343 x

5)2(3x

56x

TheoremTheorem 8.1 If the altitude is drawn to the hypotenuse of

a right triangle, then the two triangles formed are similar to the original triangle and to each other.

Write a similarity statement identifying the three similar right triangles in the

figure.

The three triangles:

FJG ~ GJH ~ FGH

Write a similarity statement identifying the three similar right triangles in the

figure.

KML ~ MPL ~ KPM

STR ~ QTS ~ QSR

TheoremGeometric Mean (Altitude) TheoremThe altitude drawn to the hypotenuse of a

right triangle separates the hypotenuse into two segments

TheoremGeometric Mean (Leg) Theorem The altitude drawn to the hypotenuse of a right triangle

separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the the length of the the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Find x, y, and z.

5

20 x

x

100xx = 10

20

25 z

z

(leg theorem)

500z 4.22510

5

25 y

y

125y

2.1155

Find x, y, & z

25

8 z

z

200z1.14210 z

25

33 y

y

7.28335 y825y

8

33 x

x

264x

2.16662 x

Find x, y, and z.

9

12

12

x

9x = 144

x = 169

25 z

z

16

225zz = 15

16

25 y

y

400yy = 20

Zach wants to order a banner that will hang over the side of his high school baseball stadium grandstand and reach the ground. To find this height, he uses a cardboard square to line up the top and bottom of the grandstand. He measures his distance from the grandstand and from the ground to his eye level. Find the height of the grandstand to the nearest foot.

x

5.10

5.10

75.5

5.75x = 110.2517.19x

Grandstand = 19.17 + 5

The grandstand is about 25 feet tall.

Homework

Pg. 535 1 – 7 all, 8 – 24 E, 28 – 36 EOE, 50, 54 – 74 EOE