Question

In the Figure,
$\left(i\right)$ $AB=BC,M$ is the mid-point of $AB$ and $N$ is the mid-point of $BC$. Show that $AM=NC$.

Answer 1

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


According to Euclid’s axiom, things which are halves of the same things are equal to one another.
From Eq. $\left(i\right),AB=BC$
On multiplying both side by $\frac{1}{2}$, we get
$\frac{1}{2}AB=\frac{1}{2}BC$
$\therefore AM=NC$ [using Eqs. $\left(ii\right)$ and $\left(iii\right)$]
Hence, proved.