You might wonder here how you’re supposed to find the area of ∆QSP 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 never provide superfluous information. In fact, because
the two angle measures provided add up to 180°, you know that and are parallel. And that,
in turn, tells you that ∆RST is similar to ∆QSP, 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 = , you know that
is twice 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.
The answer is (D).
Alternative method: If you didn’t see the similar triangles, or if you didn’t know for sure how the
area of the larger triangle is related to the area of the smaller triangle, you at least could have
eliminated some answer choices based on appearances. Look at the figure and use your eyes to
compare the areas. (We call this method eyeballing.) Doesn’t it look as though the larger triangle has
more than twice as much room inside it as the smaller triangle? That means that answer choices (A),
(B), and (C) are all visibly too small. If you can narrow the choices down to two, it certainly pays to
guess.
EYEBALL THE FIGURE
(C) c
(D) 2 c
(E) 3 c