Width of Alley

In a narrow, sinister 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, a distance k above the ground, the ladder makes a 45° angle with the ground. Resting against the other wall at R, a distance h above the ground, the ladder makes a 75° angle with the ground. The width of the alley is equal to

1. a
2. RQ
3. k
4. (h + k)/2
5. h

Show/Hide Solution

1. Draw QE ⊥ QB and AR.
2. Draw QR; △PQR is isosceles.
3. ∠RPQ = 180° – 75° – 45° = 60° → △PQR is equilateral → ∠PRQ = 60°.
4. ∠ARP = 90° – 75° = 15° → ∠EQR = 90° – (60° + 15°) = 15°.
5. △ERQ ≅ △APR, by ASA (∠ERQ = 75°).
6. h = AR = EQ = AB = w. The answer is (e).