SAT Mc Graw Hill 2011

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