Question

State true or false.

In triangle ABC , M and N are the midpoints of AB and AC respectively. The altitude AP to BC intersect at MN at O then AO = OP.

• A. True
• B. False

Answer 1

The correct option is A True

Here, In $△ABC,$

$M$ and $N$ are the midpoints of the sides $AB$ and $AC$ respectively.

Let $AP$ be the altitude from the vertex $A$ to the side $BC.$

The line joining the midpoints of $AB$ and $AC$ meet the altitude $AP$ at $O.$

$\Rightarrow$ $AM=\frac{1}{2}AB$ and $AN=\frac{1}{2}AC$

According to the mid-point theorem,

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

So, $△ABC\sim △AMN$

$\Rightarrow$ $\frac{ar\left(△AMN\right)}{ar\left(△ABC\right)}=\frac{M{N}^2}{B{C}^2}$

$\Rightarrow$ $\frac{ar\left(△AMN\right)}{ar\left(△ABC\right)}=\frac{{\left(\frac{BC}{2}\right)}^2}{B{C}^2}$

$\Rightarrow$ $\frac{ar\left(△AMN\right)}{ar\left(△ABC\right)}=\frac{1}{4}$

$\Rightarrow$ $\frac{A{O}^2}{A{P}^2}=\frac{1}{4}$

$\Rightarrow$ $\frac{AO}{AP}=\frac{1}{2}$

$\Rightarrow$ $AO=\frac{1}{2}AP$

$\Rightarrow$ $AO=OP$