Question

In Fig. 7.145, PA, QB and RC are each perpendicular to AC. Prove that $\frac{1}{x}+\frac{1}{z}=\frac{1}{y}$

Answer 1

In ΔPAC and ΔQBC
∠PCA = ∠QCB
∠PAC = ∠QBC

ΔPAC congurent to ΔQBC
PA/QB = AC/BC
x/y = AB/BC
y/x = BC/AC

In ΔRCA and ΔQBA
∠RAC = ∠QAB
∠RCA = ∠QBA
ΔRCA is congruent to ΔQBA
RC/QB = AC/AB

z/y= AC/AB
y/z= AB/AC

adding both eq
y/z + y/z = BC + AC/ AC = 1

y/z + y/z = 1

multiplying both sides by y

1/x + 1/z = 1/y