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

In the figure given below, $A B C D$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of $A B, B C, C D$ and $D A$ , respectively. $A C$ is its diagonal. Show that $P Q R S$ is a parallelogram.

Accepted Answer

In the given figure, $A B C D$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of $A B, B C, C D$ and $D A$ , respectively. $A C$ is it's diagonal. We have to show that $P Q R S$ is a parallelogram.

Construction: Draw $C X ∥ A D$ from the point $C$ and extend the line $S R$ , such that it intersects $\vec{C X}$ at the point $T$ .

Proof:

In $△ D S R$ and $△ C R T$ , we have, $∠ D R S = ∠ C R T$ [The vertically opposite angles are equal]

$R D = R C$ [ $R$ is the mid-point of $C D$ ]

$∠ S D R = ∠ R C T$ [ $A D ∥ C X$ and $C D$ is the transversal. So, they are alternate interior angles and hence, equal]

So, we have $△ D S R ≅ △ C R T$ [By ASA congruency criterion of triangles]

Thus, we have, $S R = R T$ and $D S = C T$ [CPCT] -----(i)

Now, $D S = A S$ [ $S$ is mid-point of $A D$ ] and $D S = C T$ implies $A S = C T$ .

Now, in quadrilateral $A C T S$ , we have, $A S = C T$ [From (i) and $A S ∥ C T$ [By construction]

Hence, $A C T S$ is a parallelogram.

So, $A C = S T$ and $A C ∥ S T$ [The opposite sides of a parallelogram are equal and parallel to each other]

$\Rightarrow A C = 2 S R [∵ S R = R T]$ [From (i)] and $A C ∥ S R$ [They are the same line]

Thus, we have, $S R = \frac{1}{2} A C$ and $S R ∥ A C$ . -----(ii)

[This is the mid-point theorem which can now be stated as "a line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it".]

Similarly, we can prove that $P Q = \frac{1}{2} A C$ and $P Q ∥ A C$ . -----(iii)

From (ii) and (iii), we obtain $P Q = S R$ and $P Q ∥ S R$ .

So, $P Q R S$ is a parallelogram. [If one pair of opposite sides of a quadrilateral is parallel and equal, then the quadrilateral is a parallelogram]

Hence, proved.

In the figure given below, $A B C D$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of $A B, B C, C D$ and $D A$ , respectively. $A C$ is its diagonal. Show that $P Q R S$ is a parallelogram.

Important Questions on Quadrilaterals

Learn and Prepare for any exam you want !

In the figure given below, ABCD is a quadrilateral in which P, Q, R and S are mid-points of AB, BC, CD and DA, respectively. AC is its diagonal. Show that PQRS is a parallelogram.

Important Questions on Quadrilaterals

Learn and Prepare for any exam you want !

In the figure given below, $A B C D$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of $A B, B C, C D$ and $D A$ , respectively. $A C$ is its diagonal. Show that $P Q R S$ is a parallelogram.