Vemuri wrote:Find the angle between the diagonal of a rectangle with perimeter 2p and area (3/16)p^2.
I am not sure how to determine the angle between diagonals. Any insights? I seem to be missing a concept here.
The length and the width here should be 3p/4 and p/4, which you can confirm are correct:
Area = (p/4)(3p/4) = 3p^2/16
Perimeter = 2(3p/4) + 2(p/4) = 2p
So we have a rectangle with sides in a 3 to 1 ratio. To find the angle that a diagonal makes with, say, one of the rectangle's sides (which we could then use to find the angle between the two diagonals), we'd need to know the angles in a right triangle with legs in a 3 to 1 ratio. That's not one of the special triangles you might know about, so unless I'm missing a trick, you can't easily do this without using trigonometry -- not something the GMAT ever tests. Incidentally, you absolutely should not divide the angle into a 3 to 1 ratio - angles don't work that way, which is why you spend a year studying trigonometry in high school. You can look at, for example, the 30-60-90 triangle to confirm that it's incorrect to treat angles in this way.
In any case, using a trigonometric calculator, I find that the angles in such a right triangle are 71.56... degrees and 18.43... degrees, which can then be used to determine that the smaller angle formed at the intersection of the diagonals is 36.86... degrees. Not sure how you could do that without a calculator, however. Where is the original question from?
).
