Download Geometry Notes G.7 Triangle Ratios, Proportions, Geometric Mean

Geometry Notes G.7 Triangle Ratios, Proportions, Geometric Mean
Mrs. Grieser
Name: __________________________________________ Date: ________________ Block: _______
Review:
Ratio
the comparison of two numbers using division, written “a to b”, a:b, or
Proportion
an equation that states two ratios are equal; ex:
Cross Products
a
(b ≠ 0)
b
a c
 , b  0, d  0
b d
the product of the means or extremes (see diagram below); cross products are =

Ratios should be expressed in simplest form, and can be used to find missing
dimensions

Extended ratios compare more than two numbers; e.g. ratio of sides of a triangle are
4:5:8.
o Example: Find the sides of a ∆ with the above ratio if the perimeter is 204.

Proportions are made up of parts called means and extremes:

The Cross Product Property may be used to solve proportions:
Cross Product Property
In a proportion, the product of the extremes equals the product of the means:
a c
If  , where b ≠ 0 and d ≠ 0, then ad = bc
b d
Examples:
a) The perimeter of a rectangle is 484. The ratio
of length to width is 9:2. Find the area of
the rectangle.
Solution:
Perimeter = __________
c) Solve:
5
x

10 16
b) The measures of the angles of a triangle are
in the ratio of 1:2:3. Find the measures of
the angles.
Solution:
Sketch ∆ and write in angles
according to given ratio.
d) Solve:
1
2

y  1 3y
Set up cross product: ________________
Set up cross product: ________________
Solve: ____________________________
Solve:____________________________
Geometry Notes G.7 Triangle Ratios, Proportions, Geometric Mean
Mrs. Grieser Page 2
Geometric Mean
The Geometric Mean (Mean Proportional) of two positive numbers a and b is the positive
a x

number x that satisfies:
 x2  ab  x  ab
x b
Examples:
a) Find the geometric mean of 2 and 8.
Solution:

b) Find the geometric mean of 24 and 48.
Solution:
Other proportion properties:
o Reciprocal Property: If two ratios are equal, then their reciprocals are equal:
a c
b d
if  then 
b d
a c
o If you interchange the means of a proportion, then you form another true proportion:
a c
a b
if  then 
b d
c d
o In a proportion, if you add the value of the denominator to the numerator, then you
a c
ab cd

form another true proportion:
if  , then
b d
b
d
Examples:
a b
a ?
 then 
a) If
x 5
b ?
c) T or F: If
b) If
y 7
x 10


then
x 10
y 7
x3 ?
x 9

 then
3
?
3 y
d) T or F: If
7 y
x 10


then
x 10
y 7
e) In the diagram below,
RU RV

. Find x.
US VT
f)
g) In the diagram below,
JM KN

.
ML NL
h) A highway on a map is 9 inches long. The
actual highway is 36 miles long. What is the
scale of the map?
Find JL and JM.
In the diagram below,
RU RV

. Find x.
US VT