Question

In the Figure, $BM=BN,M$ is the mid-point of $AB$ and $N$ is the mid-point of $BC$. Show that $AB=BC$

Answer 1

Prove $AB=BC$
Given: $BM=BN$ $\dots \left(i\right)$
$M$ is the mid-point of $AB$
$\therefore AM=BM$
$=\frac{1}{2}AB$
$\Rightarrow 2BM=AB$ $\dots \left(ii\right)$
and $N$ is the mid-point of $BC$

$\therefore BN=NC$
$=\frac{1}{2}BC$
$\Rightarrow 2BN=BC$ $\dots \left(iii\right)$

According to Euclid’s axiom, things which are double of the same things are equal to one another.

On multiplying both side of Eq. $\left(i\right)$ by $2$, we get:
$2BM=2BN$
$\therefore AB=BC$ [From Equation $\left(ii\right)$ and $\left(iii\right)$]
Hence, proved.