Name: __________________________
Date: ______________ Pd: ______
8-1 Geometric Mean and Pythagorean Theorem
Triangle Inequality Review: _______________________________________________________
______________________________________________________________________________
Converse of the Pythagorean Theorem
REMEMBER:
For RIGHT triangles:____________________________________
For ACUTE triangles:____________________________________
For OBTUSE triangles:___________________________________
Determine if the set of numbers can be the sides of a triangle. If yes, classify the triangle.
1.
4 cm, 5 cm, 3 cm
2.
6 in, 9 in, 7 in
3.
15 ft, 30 ft, 15 ft
4.
10 m, 10 m, 10 m
5.
8 yd, 12 yd, 8 yd
6.
50°, 50°, 80°
7.
20°, 90°, 60°
8.
150°, 20°, 10°
Ratios & Proportions: Express the following as a ratio (fraction) in simplest form.
1.
24 to 42
2.
50 minutes to 2 hours
3.
80 centimeters to 3 meters
4.
10 weeks to 35 days
A
5.
AC
AB
6.
CD
BD
7.
AD
BD
8.
AC
BC
Find AC and AD first!
___
10
D
____
3.6
4.8
C
6
B
To solve a proportion, ___________________________________________________________.
9.
2 6

3 x
13.
x3 x3

2
5
10.
10 6

6
x
14.
3x  2 2x  7

4
3
11.
9
x

x 16
15.
12.
8 x5

7 x 2
8x  12.8 2x  8

3
2.5
Geometric Mean
Formula:
Find the geometric mean of each pair of numbers.
________1. 4 and 9
________2. 5 and 10
________3. 3 and 15
Altitudes in Right Triangles
When a right triangle has an altitude, the segment lengths have special ratios.
RULE #1:
The altitude is the geometric mean of the two segments of the hypotenuse.
Altitude:
B
BD
AD and DC
Hypotenuse Segments:
D
A
Formula:
C
AD BD
(remember the formula for geometric mean!)

BD DC
Use the formula and triangle from above to solve the following problems.
10 x

x 2
1. Suppose AD = 10 and DC = 2. Find BD.
x2 = 20
x  20  4 5  2 5
2. Suppose BD = 10 and AD = 25. Find DC.
_____ = _____
DC = ____________
3. Using the answer for #2, find AC. (Hint: AC = ? + ?)
AC = ____________
4. Suppose AC = 12 and DC = 4. Find AD first. Then use the formula to find BD.
AD = ____________
_____ = _____
BD = ____________
Write the altitude formula (geometric mean) for the following triangles.
F
5.
E
6.
H
G
W
M
Z
X
J
7.
Y
L
_____ = _____
_____ = _____
_____ = _____
K
RULE #2:
A leg of the right triangle is the geometric mean between the whole hypotenuse and the
part of the hypotenuse adjacent to that leg.
B
Leg = AB
Whole Hypotenuse: AC
D
A
C
Adjacent Part of Hypotenuse: AD
Formula:
AC AB

AB AD
Note: If BC is used instead of
AB , then DC would replace AD .
Use the formula and triangle from above to solve the following problems.
8. Suppose AC = 12 and DC = 2. Find AD first. Then find AB.
12 x

x  120  4 30  2 30
AD = 12 – 2 = 10
x2 = 120
x 10
9. Suppose AB = 12 and AD = 9. Find AC.
_____ = _____
AC = ____________
10. Using the answer from #9, find DC.
DC = ____________
11. Suppose BC = 8 and DC = 4. Find AC first. Then find AD.
_____ = _____
AC = ____________
AD = ____________
12. Write the geometric mean formulas for each leg of ∆EFG.
F
Leg = EF
Leg = FG
_____ = _____
_____ = _____
E
H
G