Show that the quadrilateral formed by …

In a square A B C D , P , Q , R and S are the mid-points of A B , B C , C D and D A respectively. ⇒ A B = B C = C D = A D [ Sides of square are equal ] In △ A D C , S R ∥ A C and S R = 1 2 A C [ By mid-point theorem ] ---- ( 1 ) In △ A B C , P Q ∥ A C and P Q = 1 2 A C [ By mid-point theorem ] ---- ( 2 ) From equation ( 1 ) and ( 2 ), S R ∥ P Q and S R = P Q = 1 2 A C ---- ( 3 ) Similarly, S P ∥ B D and B D ∥ R Q ∴ S P ∥ R Q and S P = 1 2 B D and R Q = 1 2 B D ∴ S P = R Q = 1 2 B D Since, diagonals of a square bisect each other at right angle. ∴ A C = B D ⇒ S P = R Q = 1 2 A C ----- ( 4 ) From ( 3 ) and ( 4 ) S R = P Q = S P = R Q We know that the diagonals of a square bisect each other at right angles. ∠ E O F = 90 o . Now, R Q ∥ D B R E ∥ F O Also, S R ∥ A C ⇒ F R ∥ O E ∴ O E R F is a parallelogram. So, ∠ F R E = ∠ E O F = 90 o (Opposite angles are equal) Thus, P Q R S is a parallelogram with ∠ R = 90 o and S R = P Q = S P = R Q ∴ P Q R S is a square.