Question

In the given figure, it is given that ZABD = ZCDB ZPQB = 90°. If AB = x units, CD = y units and 1 1 1 PQ = z units, prove that 1/x+1/y=1/z​

Answer 1

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In ΔABD and ΔPQD

∠D= ∠D   (common)

∠BAD=∠QPD   (corresponding angles)

So, ΔABD∼ΔPQD     (by AA similarity criterion)

⇒PQAB=QDBD  (Ratio of corresponding sides of two similar triangles)

⇒zx=QDBQ+QD

⇒zx=QDBQ+1

zx−1=QDBQ

⇒zx−z=QDBQ    ---(i)

Similarly in ΔCBD and ΔPBQ

∠B=∠B  (common)

∠BCD=∠BPQ  (corresponding angles)

So,ΔCBD∼ΔPBQ  (by AA similarity criterion)

⇒PQCD=BQBD  (Ratio of corresponding sides of two similar triangles)

⇒zy=