Question

in the fig AB//PQ//CD, AB=x units ,CD= y units ,PQ=z units. prove that 1/x +1/y=1/z

Answer 1

Lets take BQ = a
and
DQ = b

In the shown diagram,
AB|| PQ ||CD

Therefore,

In ΔABD and ΔDPQ

∠DPQ = ∠DAB (corresponding angle)
∠DQP = ∠DBA (corresponding angle)

Therefore,
ΔABD ~ ΔDPQ

Now,

PQ/AB = z/x = a/(a+b) --(i) (the property of similar triangles)

Similarly,

ΔCBD ~ Δ ΒPQ

and
PQ/CD = z/y = (a)/( a+b) ---(ii)

Adding (i) and (ii)

$\frac{z}{x} + \frac{z}{y} = \frac{b}{a + b} + \frac{a}{a + b} \\ z( \frac{1}{x} + \frac{1}{y} ) = 1 \\ \frac{1}{x} + \frac{1}{y} = \frac{1}{z}$


Hence proved.