1982 AHSME Problems/Problem 22

Problem 22

In a narrow alley of width $w$ a ladder of length $a$ is placed with its foot at point P between the walls. Resting against one wall at $Q$ , the distance k above the ground makes a $45^\circ$ angle with the ground. Resting against the other wall at $R$ , a distance h above the ground, the ladder makes a $75^\circ$ angle with the ground. The width $w$ is equal to

$\text{(A)}a\qquad \text{(B)}RQ\qquad \text{(C)}k\qquad \text{(D)}\frac{h+k}{2}\qquad \text{(E)}h$

Solution

$[asy] import olympiad; unitsize(40); pair T,P,Q,M,L,R; T=(0,0); P=(2.5,0); Q=(10,7.5); M=(0,7.5); L=(10,0); R=(0,10); draw(R--T--L--Q); draw(P--Q--R--cycle); draw(Q--M); label("$P$", P, S); dot(P); label("$Q$", Q, E); dot(Q); label("$R$", R, W); dot(R); label("$S$", M, W); dot(M); label("$T$", T, S); dot(T); label("$L$", L, S); dot(L); label("$w$",(5,-1)); label("$h$",(-1,5)); markscalefactor=0.03; draw(anglemark(L,P,Q)); draw(anglemark(R,P,T)); draw(rightanglemark(P,T,M)); draw(rightanglemark(Q,L,P)); [/asy]$

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1982 AHSME Problems/Problem 22

Problem 22

Solution