3155.51 – A Narrow Sinister Alley

In the alley, a ladder of length $a$ is placed with its foot at the point P (see the figure). This ladder rests against one wall at the point Q, at a height of $k$ above the ground and makes an angle of 45 degrees with the alley. A second ladder of the same length rests against the other wall at the point R at a height $h$ above the ground. It makes an angle of 75 degrees with the alley. Which of these quantities represents the width $w$ of the alley?

$a$
RQ
$k$
$\frac{h+k}{2}$
$h$

Solution

OUTLINE:

To the given figure, add the line RQ and drop a perpendicular from R to X, on the side of the alley on which Q sits. Let the new length XQ be $z$ .
Justify the angle measurements in the new figure.
Justify the length measurements in the new figure.
Explain why $w = z+k = h.$

Thus the correct answer is e.