Question

If $P,Q$ and $R$ are the mid-points of the sides $BC,CA$ and $AB$ of a triangle and $AD$ is the perpendicular from $A$ on $BC$ then $P,Q,R$ and $D$ are concyclic.

If the statement is true then answer $0$ and if the statement is false then answer $1$

Answer 1

Given:In $△ABCP,Q$ and $R$ are the midpoints of the sides

$BC,CA$ and $AB$ respectively.

Also,$AD\perp BC$

In a right-angled triangle, $ADP,R$ is the midpoint of $AB$

$\therefore RB=RD$

$\Rightarrow \angle 2=\angle 1$ ........$\left(1\right)$

since angles opposite to the equal sides are equal.

Since,$R$ and $Q$ are the mid-points of $AB$ and $AC$, then $RQ\parallel BC$

or $RQ\parallel BP$ (by mid-point theorem)

Since,$QP\parallel RB$ then quadrilateral $BPQR$ is a parallelogram,

$\Rightarrow \angle 1=\angle 3$ ........$\left(2\right)$

since angles opposite to the equal sides are equal.

From equations $\left(1\right)$ and $\left(2\right)$,

$\angle 2=\angle 3$

But $\angle 2+\angle 4={180}^{\circ }$ (by linear pair axiom)

$\angle 3+\angle 4={180}^{\circ }(\because \angle 2=\angle 3)$

Hence ,quadrilateral $PQRD$ is a cyclic quadrilateral.

So,points $P,Q,R$ and $D$ are con-cyclic.

Hence the statement is true.

The answer is $0$