Question

ABCD is arectangle and P,Q,R, and S are mind points of the sidesAB,BC,CD and DA respectively.Show that the quadrilateral PQRS is arhombus.

Answer 1

$\huge\bold{\underline{Question:}}$

ABCD is arectangle and P,Q,R, and S are mind points of the sidesAB,BC,CD and DA respectively.Show that the quadrilateral PQRS is arhombus.

$\huge\bold{\underline{Answer:}}$

Given:

• ABCD is a rectangle.

• P,Q,R, and S are mind points of the sidesAB,BC,CD and DA respectively.

To prove:

Show that the quadrilateral PQRS is arhombus.

Proof:

Join DB and AC .

We know that the line joining the midpoints of the two sides of triangle is parallel to the third side and is of half the measure of third side.

Here in triangle ADB P and S are the midpoints of AB and AD respectively,hence

• PS || DB and PS = 1/2 DB.............➊

Similarly ,

• RQ || DB and RQ = 1/2 DB.............➋

• PQ || AC and PQ = 1/2 AC..............➌

• SR || AC and SR = 1/2 AC...............➍

From equation 1 and 2 we get,

$\red\bigstar$ PS || RQ and PS = RQ

And from equation 3 and 4 we get

$\blue\bigstar$ PQ || SR and PQ = SR

Therefore, PQRS is a rhombus. (Proved)