Question

$ABC$ is a right angles triangle with $\angle B={90}^{\circ },m$ is the midpoint of $AC$ and $Bm=\sqrt{117}cm,AB+BC=30$, then the area of the triangle is _______.

• A. $108c{m}^2$
• B. $248c{m}^2$
• C. $316c{m}^2$
• D. $156c{m}^2$

Answer 1

The correct option is A $108c{m}^2$

We know the length og median through B, $Bm=\frac{1}{2}\times \sqrt{2A{B}^2+2B{C}^2-A{C}^2}$

But by Pythoras theorem, $A{B}^2+B{C}^2=A{C}^2$

$\Rightarrow Bm=\frac{AC}{2}$

$\Rightarrow A{C}^2=4\times 117=468$

Now $AB+BC=30$

So, $(AB+BC{)}^2=900$

$\Rightarrow A{B}^2+A{C}^2+2AB\times AC=900$

$\Rightarrow AB\times AC=(900-A{C}^2)\times \frac{1}{2}=216$

In $△ABC$, $area=\frac{1}{2}\times AC\times BC=108$

So, Option (A)