If a : b = c : d, then show that `sqrt(a^(2) + c^(2)) : sqrt(b^(2) + d^(2)) = (pa + qc) : (pb+qd)`

Explanation

Given that ` a : b = c : d rArr a/b = c/d = k (let) [k ne 0]`
`:. ` a = bk and x = dk.
Now, LHS = `sqrt(a^(2)+c^(2)) : sqrt(b^(2) + d^(2)) = sqrt(a^(2) + c^(2))/sqrt(b^(2) + d^(2)) = sqrt((bk)^(2) + (dk)^(2))/sqrt(b^(2) + d^(2))`
`= sqrt(b^(2)k^(2)+d^(2)k^(2))/sqrt(b^(2)+d^(2)) = sqrt(k^(2)(b^(2) + d^(2)))/sqrt(b^(2) + d^(2)) = (ksqrt(b^(2)+d^(2)))/sqrt(b^(2) +d^(2) ) = k`
RHS ` = (pa + qc) : (pb + qd) = (pa+qc)/(pb+qd)`
` = ( xx bk + q xxdk)/(pb + qd) = (k(pb+qd))/((pb+qd)) = k`.
` :. ` LHS = RHS