CHAPTER 10 / ESSENTIAL GEOMETRY SKILLS 389
Similar Figures
When you think of similaryou probably think
of “almost the same, but not quite.” In mathe-
matics, however, the word similarhas a much
more specific, technical meaning. Two figures
are similar if they are the same shape, but not
necessarily the same size. For instance, all cir-
cles are similar to each other, and all squares
are similar to each other: there is only one
“shape” for a circle, and only one “shape” for a
square. But there are many different shapes
that a rectangle may have, so two rectangles
aren’t necessarily similar.
If two shapes are similar, then all correspond-
ing angles are equal and all corresponding
lengths are proportional.
Use proportions to find the lengths of un-
known sides in similar figures.
Example:
What is xin the figure at left below?
- Two pairs of corre-
sponding sides are
proportional and the
angles between them are equal. - All three pairs of
corresponding sides
are proportional.
Ratios of Areas
Consider two squares: one with a side length of 2 and
the other with a side length of 3. Clearly, their sides
are in the ratio of 2:3. What about their areas? That’s
easy: their areas are 2^2 =4 and 3^2 =9, so the areas are
in a ratio of 4:9. This demonstrates a fact that is true
of all similar figures:
If corresponding lengths of two similar figures
have a ratio of a:b, then the areas of the two fig-
ures have a ratio of a^2 :b^2.
Example:
A garden that is 30 feet long has an area of 600
square feet. A blueprint of the garden that is
drawn to scale depicts the garden as being 3 inches
long. What is the area of the blueprint drawing of
the garden?
It is tempting to want to say 60 square inches be-
cause 30:600 =3:60. But be careful: the ratio of
areas is the square of the ratio of lengths! You can
draw a diagram, assuming the garden to be a rec-
tangle. (The shape of the garden doesn’t matter: it’s
convenient to draw the garden as a rectangle, but
it doesn’t have to be.) Or you can simply set up the
proportion using the formula:
Cross-multiply: 900 x=5,400
Divide by 900: x= 6
x
600
3
30
9
900
2
== 2
Lesson 6: Similar Figures
a°
a°
b°
b°
10 7
8
x
The two triangles are similar because all of their corre-
sponding angles are equal. (Even though only two
pairs of angles are givenas equal, we know that the
other pair are also equal, because the angles in a trian-
gle must add up to 180°.) So we can set up a proportion
of corresponding sides:
Cross-multiply: 7x= 80
Divide by 7: x=80/7 =11.43
Two triangles are similar if any of the follow-
ing is true:
- Two pairs of corre-
sponding angles are
equal. (If two pairs
are equal, the third pair must be equal, too.)
10
78
=
x
30 feet
20 feet
3 inches
a° a° 600 sq. ft 2 inches 6 sq. in
b° b°
a° a°
(^42)
6 3
(^42)
6 3
(^3) 1.5