SIMILAR FIGURES
The area ratio between similar figures is the square of the side ratio.You might wonder how you’re supposed to find the area of ∆ACD when you’re
given no lengths you can use for a base or an altitude. The only numbers you
have are the angle measures. They must be there for some reason—the test
makers rarely provide superfluous information. In fact, because the two angle
measures provided add up to 180°, they tell you that and are
parallel. And that, in turn, tells you that ∆ABE is similar to ∆ACD—because they
have the same three angles.
Similar triangles are triangles that have the same shape: Corresponding
angles are equal, and corresponding sides are proportional. In this case,
because it’s given that AB = BC, you know that AC is twice AB and that
corresponding sides are in a ratio of 2:1. Each side of the larger triangle is
twice the length of the corresponding side of the smaller triangle. That
doesn’t mean, however, that the ratio of the areas is also 2:1. In fact, the area
ratio is the square of the side ratio, and the larger triangle has four times the
area of the smaller triangle, so the answer is (E).
(C) 2 x
(D) 3 x
(E) 4 x