Question

In Fig. 4.142, PA, QB and RC are each perpendicular to AC. Prove that 1/x + 1/z = 1/y.

Answer 1

In Fig. 4.142, PA, QB and RC are each perpendicular to AC.

Hence, it is proved that 1/x + 1/z = 1/y.

Given,

PA, QB and RC are each perpendicular to AC

In Δ ABQ and Δ ACR

∠ BAQ = ∠ CAR  [common angles]

∠ ABQ = ∠ ACR = 90°

Therefore, by AA criteria,

Δ ABQ ~ Δ ACR

⇒ AB/AC = BQ/CR ...........(1)

In Δ CBQ and Δ CAP

∠ BCQ = ∠ ACP  [common angles]

∠ CBQ = ∠ CAP = 90°

Therefore, by AA criteria,

Δ CBQ ~ Δ CAP

⇒ CB/CA = BQ/AP ...........(2)

Adding equations (1) and (2), we get,

AB/AC +CB/CA  = BQ/CR + BQ/AP

(CA+AB) / AC = y/x + y/z

AC / AC = y (1/x + 1/z)

1/y = 1/x + 1/z

Hence is the proof.