Question

If a, b, c, d are in proportion, prove that (a² + c²) : (b² + d²) = ac : bd

Answer 1

`a/b = c/d` = k

a = bk and c = dk

L.H.S.

= `(a^2 + c^2)/(b^2 + d^2)`

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

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

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

= k²

R.H.S.

= `(ac)/(bd)`

= `((bk)(dk))/(bd)`

= `(bdk^2)/(bd)`

= `(bdk^2)/(bd)`

= k²

L.H.S. = R.H.S.