# 1982 AHSME Problems/Problem 18

Revision as of 07:42, 15 April 2016 by Katzrockso (talk | contribs) (Created page with "== Problem 18 == In the adjoining figure of a rectangular solid, <math>\angle DHG=45^\circ</math> and <math>\angle FHB=60^\circ</math>. Find the cosine of <math>\angle BHD</m...")

## Problem 18

In the adjoining figure of a rectangular solid, and . Find the cosine of .

## Solution

WLOG, let .

Looking at square GHDC, we see that , which implies that and

Taking each cross-section one at a time, we look at square DHFB. We obviously know that CHB is a degree angle, giving , and .

Looking at square ABCD, we see that .

We now look at triangle DBH, which has side lengths , and . Because is opposite angle DHB, by the law of cosines,

Thus, the answer is