Question

PA,QB and RCare perpendicular to AC .
PA=a
QB=y
RC=z
Then prove that 1/x+1/z=1/y

Answer 1

Solution:-

In Δ PAC and Δ QBC,

∠ PCA = ∠ QCB       (Common angle)

∠ PAC = ∠ QBC       (Right angle each)

⇒ Δ PAC ~ Δ QBC    (AA Similarity)

⇒ PA/QB = AC/BC    (Similar triangles have proportional sides)

⇒ x/y = AC/BC ⇒ y/x = BC/AC .....(1)

In Δ RCA and Δ QBA,

∠ RAC = ∠ QAB        (Common angle)

∠ RCA = ∠ QBA        (Right angle each)

⇒ Δ RCA ~ Δ QBA     (AA Similarity)

⇒ RC/QB = AC/AB 

⇒ z/y = AC/AB ⇒ y/z = AB/AC ......(2)

Adding (1) and (2), we get

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

⇒ y/x + y/z = 1

On multiplying both sides by 1/y, we get

1/x + 1/z = 1/y

Hence proved.

Answer 2

Hope this helps you