Question

In $\Delta ABC$, M and N are the mid-points of AB and AC respectively. The altitude AP to BC interest MN at O.

Prove that $AO=OP$

Answer 1

Given:- $ABC$ is a $△,M$ and $N$ are the mid-points of $AB$ and $AC$ respectively. The altitude $AP$ to $BC$ intersect $MN$ at $O$.

To prove:- $AO=OP$

Proof:-

Since $M$ and $N$ are the mid points of $AB$ and $BC$,

By mid point Theorem,

$MN\parallel BC$

$MN=\frac{1}{2}BC$

Now In $△ABP$,

Since $M$ is the mid point of $AB,MO\parallel BP$.

Therefore by Converse of midpoint theorem

$O$ is the midpoint of $AP$

$\therefore AO=OP$

Hence proved.