`bar(QR)=bar(RS)`. If the area of `Delta RST` is `(c )/(2)`, what is the area of `Delta QSP` ?

Options

A. `c sqrt(2)`
B. `c sqrt(3)`
C. c
D. 2c

Answer

D

Explanation

You might wonder here how you're supposed to find the area of `Delta QSP` when you're given no lengths you can for a base or an altitude. The only numbers you have are the angle measures. They must be there for some reason - the makers never provide superfluous information. In fact, because the two angle measures provided add up to `180^(@)`, you know that `bar(PQ)` and `bar(TR)` are parallel. And that, in turn you that `Delta RST` is similar to `Delta 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 `bar(QR)=bar(RS)`, you know that `bar(QS)` is twice `bar(QR)` 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.
area of `Delta QSP=4xx` area of `Delta RST=4xx(c )/(2)=2c`.