Question

If the sides of a right triangle are $9,12$ and $15$ cm long then the sum of squares of medians is:

• A. $227.5$
• B. $337.5$
• C. $537.5$
• D. None of these

Answer 1

The correct option is B $337.5$

$A{D}^2+B{E}^2+C{F}^2=(\frac{A{B}^2+A{C}^2}{2}-\frac{B{D}^2}{4})+(\frac{A{B}^2+B{C}^2}{2}-\frac{A{E}^2}{4})+(\frac{A{C}^2+B{C}^2}{2}-\frac{A{F}^2}{4})$

$A{D}^2+B{E}^2+C{F}^2=[\frac{2(A{B}^2+B{C}^2+C{A}^2}{2}-\frac{(B{D}^2+A{E}^2+A{F}^2)}{4}]$

$A{D}^2+B{E}^2+C{F}^2=(9{)}^2+(12{)}^2+(15{)}^2-\frac{(6{)}^2+(4.5{)}^2+(1.5{)}^2}{4}$

$=81+144+225-\left(\frac{36+20.25+56.25}{4}\right)=450-112.5=337.5$

Hence, $337.5$ is the correct answer.