In triangle ABC,AB=2cm BC=3cm CA=4cm.D is the midpoint of AC.if a square is constructed on the side BD,what is the area of the square

Question

In triangle $\Delta ABC$, $AB=2\text{ cm}$, $BC=3\text{ cm}$, $CA=4\text{ cm}$. $D$ is the midpoint of $AC$. If a square is constructed on the side $BD$,what is the area of the square. Now i used similarity between $\Delta ABD$ and $\Delta BCD$ but to no result..

Answer 1

All you need is to find the length of the median $BD$. You can derive the following median formula using Cosine Law.

$$ \begin{align} &M_b=\frac{1}{2}\sqrt{2c^2+2a^2-b^2} \\\Rightarrow\ &BD=\frac{\sqrt{10}}{2} \\\Rightarrow\ &{\square}_{Area}=\frac{5}{2} \end{align} $$