18.79 As shown in the figure, the distance between two walls is $w$, and there is a ladder at point $P$ between them. The length of the ladder is $a$. The ladder leans against one wall, with its upper end touching the wall at point $Q$. The distance from $Q$ to the ground is $k$. At this moment, the ladder forms a $45^{\circ}$ angle with the ground. The ladder then leans against the other wall, with its upper end touching the wall at point $R$. The distance from $R$ to the ground is $h$, and at this moment, the ladder forms a $75^{\circ}$ angle with the ground. What is the distance $w$ between the two walls?
(A) $a$.
(B) $R Q$.
(C) $k$.
(D) $\frac{h+k}{2}$.
(E) $h$.
[Solution] As shown in the figure, draw $R X \perp B Q$, intersecting the extension of $B Q$ at $X$.

Since $\angle Q P R=60^{\circ}$, $\triangle P R Q$ is an equilateral triangle, i.e., $R Q=a$.
At the same time, $\angle Q R X=90^{\circ}-15^{\circ}-60^{\circ}=15^{\circ}$
$$
=\angle A R P \text {, }
$$

Therefore, $\triangle A R P \cong \triangle X R Q$, so $w=h$. Hence, the answer is $(E)$.