Question

In $\Delta$ABC, m$\angle$B $=90$, $\overline{BM}\perp \overline{AC}$, $M\in \overline{AC}$. If AM$-$MC $=7$, $A{B}^2-B{C}^2=175$. Find AC.

Answer 1

Given $\angle BMC=0,\angle ABC={90}^{\circ }$

$AM-MC=7$

$A{B}^2-B{C}^2=175$

In $\Delta AMB$ & $\Delta CMB$

using pythagoras theorem

$A{B}^2=A{M}^2+M{B}^2$ ___(1)

$A{C}^2=M{C}^2+B{M}^2$ __(2)

equation (1) - equation (2)

$A{B}^2-A{C}^2=A{M}^2-M{C}^2=175$

$\Rightarrow (AM-AC)(AM+MC)=175$

$\Rightarrow 7AC=175$

$AC=25$