ABCD is a rectangle formed by 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 AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square ? a rectangle ? or a rhombus ? Justify your answer.

We have, following given points

Fig. 7.23.
A(-1, -1)
B(-1, 4)
C(5, 4)
And D (5, -1)
Therefore, co-ordinates of P are

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Co-ordinates of Q are

equals straight Q open parentheses fraction numerator negative 1 plus 5 over denominator 2 end fraction comma fraction numerator 9 plus 4 over denominator 2 end fraction close parentheses rightwards double arrow straight Q space left parenthesis 2 comma space 4 right parenthesis

Co-ordinates of R are

equals straight R open parentheses fraction numerator 5 plus 5 over denominator 2 end fraction comma fraction numerator negative 1 plus 4 over denominator 2 end fraction close parentheses rightwards double arrow straight R open parentheses 5 comma 3 over 2 close parentheses

And, co-ordinates of S are

equals straight S open parentheses fraction numerator negative 1 plus 5 over denominator 2 end fraction comma fraction numerator negative 1 minus 1 over denominator 2 end fraction close parentheses rightwards double arrow straight S space left parenthesis 2 comma space 1 right parenthesis

Now,

Wee see that
PR = QR = RS = SP
and PR ≠ QS
Therefore PQRS is a rhombus.

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