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Question

# ABCD is a rectangle formed by joining the points A (−1, −1), B(−1 4) C (5 4) and D (5, −1). P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

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Solution

## We have a rectangle ABCD formed by joining the points A (−1,−1); B (−1, 4); C (5, 4) and D (5,−1). The mid-points of the sides AB, BC, CD and DA are P, Q, R, S respectively. We have to find that whether PQRS is a square, rectangle or rhombus. In general to find the mid-point of two pointsand we use section formula as, Therefore mid-point P of side AB can be written as, Now equate the individual terms to get, So co-ordinates of P is Similarly mid-point Q of side BC can be written as, Now equate the individual terms to get, So co-ordinates of Q is (2, 4) Similarly mid-point R of side CD can be written as, Now equate the individual terms to get, So co-ordinates of R is Similarly mid-point S of side DA can be written as, Now equate the individual terms to get, So co-ordinates of S is (2,−1) So we should find the lengths of sides of quadrilateral PQRS. All the sides of quadrilateral are equal. So now we will check the lengths of the diagonals. All the sides are equal but the diagonals are unequal. Hence ABCD is a rhombus.

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