Question

If `a/b = c/d`, show that: `(a + b) : (c + d) = sqrt(a^2 + b^2) : sqrt(c^2 + d^2)`

Answer 1

Let `a/b = c/d = k`   ...(Say)

`=>` a = bk, c = dk

L.H.S = `(a + b)/(c + d)`

= `(bk + b)/(dk + d)`

= `(b(k + 1))/(d(k + 1))`

= `b/d`

R.H.S = `sqrt(a^2 + b^2)/sqrt(c^2 + d^2)`

= `sqrt((bk)^2 + b^2)/sqrt((dk)^2 + d^2)`

= `sqrt(b^2(k^2 + 1))/sqrt(d^2(k^2 + 1))`

= `sqrt(b^2)/sqrt(d^2)`

= `b/d`

∴ L.H.S = R.H.S