$P, Q, R, and S$ are respectively the mid-points of the sides $A B, B C, C D, and D A$ of a quadrilateral $A B C D$ in which $A C = B D$ . Prove that $P Q R S$ is a rhombus.

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Solution

Prove that all sides are equal and parallel.
It is given that $P, Q, R, and S$ are respectively the mid-points of the sides $A B, B C, C D, and D A$ of a quadrilateral $A B C D$ .
Apply the mid-point theorem in $∆ A B D$ and in $∆ B D C$ .
In $∆ A B D$ , $P S = \frac{1}{2} B D$ and $P S ∥ B D$ .
In $∆ B D C$ , $Q R = \frac{1}{2} B D$ and $Q R ∥ B D$ .
$\Rightarrow P S = Q R = \frac{1}{2} B D \dots (1)$ and $P S ∥ Q R$ .
Again, apply the mid-point theorem in $∆ A B C$ and in $∆ A D C$ .
In $∆ A B C$ , $P Q = \frac{1}{2} A C$ and $P Q ∥ A C$ .
In $∆ A D C$ , $S R = \frac{1}{2} A C$ and $S R ∥ A C$ .
$\Rightarrow P Q = S R = \frac{1}{2} A C$ and $P S ∥ Q R$ .
Since it is given that $A C = B D$ , $P Q = S R = \frac{1}{2} B D \dots (2)$ , from $(1)$ and $(2)$ , $P S = Q R = P Q = S R$ .
Hence, all the four sides of $P Q R S$ are equal.
Hence, $PQRS$ is a rhombus.

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