Name: Rectangles and Squares Formed by Four Points
Uploaded: 2019-07-19
Duration: 6 min 3 s
Description: What are the conditions for a quadrilateral to be a rectangle? How does it differ from a square? Unlock your doubts and clear your basics from this video!

Question

P

and

Q

are points on opposite sides

AD

and

BC

of a parallelogram

ABCD

such that

PQ

passes through the point of intersection

O

of its diagonals

AC

and

BD

. Show that

PQ

is bisected at

O

.

Accepted Answer

Given, $Q$ is a parallelogram. $P$ and $Q$ are the points on opposite sides $A D$ and $AC$ . The diagonals $AC$ and $BD$ intersect at $O$ . $P Q$ passes through the point of intersection $O$ of its diagonals.

We have to show that $P Q$ is bisected at $O$ .

Consider $△ ODP$ and $△ OBQ$ . We have,

$∠ B O Q = ∠ P O D$ [the vertically opposite angles are equal].

$∠ O B Q = ∠ O D P$ [the alternate interior angles are equal].

and, $OB = OD$ [Diagonals of a parallelogram bisect each other]

Hence, by ASA criterion of congruency of a triangle, we obtain, $△ O D P ≅ △ O B Q$ . [ASA criterion states that two triangles are congruent if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle].

So, we get, $O P = O Q$ [When two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements].

Thus, $P Q$ is bisected at $O$ .

Hence, proved.

$P$ and $Q$ are points on opposite sides $AD$ and $BC$ of a parallelogram $ABCD$ such that $PQ$ passes through the point of intersection $O$ of its diagonals $AC$ and $BD$ . Show that $PQ$ is bisected at $O$ .

Important Questions on Quadrilaterals

Learn and Prepare for any exam you want !

P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O.

Important Questions on Quadrilaterals

Learn and Prepare for any exam you want !

$P$ and $Q$ are points on opposite sides $AD$ and $BC$ of a parallelogram $ABCD$ such that $PQ$ passes through the point of intersection $O$ of its diagonals $AC$ and $BD$ . Show that $PQ$ is bisected at $O$ .