Question

In $△$ MGN, MP $\perp$ GN. If MG = a units, MN = b units, GP = c units and PN = d units, prove that $\frac{(a-b)}{(c-d)}=\frac{(c+d)}{(a+b)}$

Answer 1

In $\Delta GMP$,

Use Pythagoras Theorem,

$a^2=(MP{)}^2+c^2$

$⟹a^2-c^2=(MP{)}^2$

In $\Delta MNP$,

Use Pythagoras Theorem,

$b^2=(MP{)}^2+d^2$

$⟹b^2-d^2=(MP{)}^2$

Equating both,

$a^2-c^2=b^2-d^2$

$⟹a^2-b^2=c^2-d^2$

$⟹(a-b)(a+b)=(c-d)(c+d)$

$⟹\frac{(a-b)}{(c-d)}=\frac{(c+d)}{(a+b)}$