Question

In the given figure $△ABC$ is a right-angled at $B.$ $AD$ and $CE$ are the two medians drawn from $A$ and $C$ respectively. If $AC=5$ cm and $AD=\frac{3\sqrt{5}}{2}$ cm, then $CE$ equals

• A. $2$ CM
• B. $2\sqrt{5}$ cm
• C. $5\sqrt{2}$ cm
• D. $3\sqrt{2}$ cm

Answer 1

The correct option is B $2\sqrt{5}$ cm

ABC is a right angled triangle, with right angle at ${90}^{\circ }$
In $△ABD$, Using Pythagoras theorem
$A{B}^2+B{D}^2=A{D}^2$
$4A{B}^2+B{C}^2=4A{D}^2$ ....... $\left(1\right)$
In $△BEC$, Using Pythagoras theorem
$B{E}^2+B{C}^2=C{E}^2$
$4B{C}^2+A{B}^2=4C{E}^2$ ........ $\left(2\right)$
Adding $\left(1\right)$ and $\left(2\right),$ we get
$5A{B}^2+5B{C}^2=4(A{D}^2+C{E}^2)$
$5A{C}^2=4(A{D}^2+C{E}^2)$ ..... (Using Pythagoras theorem in triangle $ABC$)
$125=45+4C{E}^2$
$C{E}^2=20$
$\Rightarrow CE=2\sqrt{5}$ cm