Ratios of Areas
Ratio of Areas:
What is the area ratio between
ABCD and XYZ?
One way of determining the ratio of the areas of two
figures is to calculate the quotient of the two areas.
B
A
Y
10
D
9
8
C
X
Z
12
1. Ratio A
A
Steps:
1. Set up fraction
2. Write formulas
2. A
A
= b1h1
= 1/2b2h2
3. Plug in numbers
3.
= 9•10
1/ • 8 •12
2
4. Solve and label
with units
= 90
48
=15
8
Find the ratio of
D
C
B
ABD to
CBD
When AB = 5 and BC = 2
A
1. Ratio ABD
CBD
2. ABC
CBD
3.
= 1/2b1h
= 1/2b2h
= ½(5)h
½(2)h
4. 5:2 or 5/2
Similar triangles:
Ratio of any pair of corresponding ,
altitudes, medians, angle bisectors, equals
the ratio of their corresponding sides.
Given ∆ PQR
∆WXY
Find the ratio of the area.
Q
X
6
4
W
P
R
First find the ratio of the sides.
QP = 6
XW 4
=3
2
Y
Q
X
4
6
W
P
Y
R
1/ b h
A
=
Ratio of area:
PQR
2 1 1
1/ b h
AWXY
2 2 2
= b1h1
b2h2
Because they are similar
=
3•3
triangles the ratios of the
2•2
sides and heights are
=9
the same.
4
Area ratio is the sides ratio squared!
Theorem 109: If 2 figures are similar, then
the ratio of their areas equals the square
of the ratio of the corresponding
segments. (similar-figures Theorem)
A1 = S1
A2
S2
2
When A1 and A2 are areas
and S1 and S2 are measures
of corresponding segments.
Given the similar
pentagons shown, find
the ratio of their areas.
S1 = 12
S2 9
A1 = 4
A2
3
=4
3
2
= 16
9
If the ratio of the areas of two similar
parallelograms is 49:121, find the ratio of their
bases.
121 cm2
49cm2
A1 = S1
A2
S2
2
49 = S1
121
S2
2
7 = S1
11
S2
Corresponding Segments include:
Sides, altitudes, medians, diagonals, and
radii.
A
Ex. AM is the median of
∆ABC. Find the ratio of
B
M
C
A ∆ ABM : A ∆ACM
Notice these are not similar figures!
A
B
M
C
1. Altitude from A is
congruent for both
triangles. Label it X.
2. BM = MC because AM is a median.
Let y = BM and MC.
A∆ABM = 1/2 b1h1
A ∆ACM 1/2 b2h2
= 1/2 xy
1/ xy
2
=1
Therefore the ratio is 1:1 They are equal !
Theorem 110: The median of a triangle
divides the triangle into two triangles with
equal area.