question_answer

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle if

A)  PQRS is a rectangle

B)         PQRS is a parallelogram

C)         Diagonals of PQRS are equal

D)         Diagonals of PQRS are at right angles

Correct Answer: D

Solution :

Let A, B, C and D be the mid-points of PQ, QR, RS and SP respectively Now, In \[\Delta RSQ,C\]and B are the mid-points of RS and RQ respectively. So, by mid-point theorem. \[CB||SQ\]                                ?(i) Similarly, In \[\Delta PSQ,\]                     \[DA||SQ\]                                ?(ii) In \[\Delta SPR,\] \[CD||RP\]                                ?(iii) Also, in\[\Delta QRP\] \[AB||RP\]                                ?(iv) From (i) and (ii), \[CB||DA\]                                ?(v) From (iii)and (iv), \[CD||AB\] Hence, from (v) and (vi), ABCD is a parallelogram. Now, if diagonals bisect SQ and PR are at \[{{90}^{o}}\] Then, \[CB\bot CD,\,CB\bot AB,\,AB\bot DA\]and \[AD\bot CD.\] So, ABCD is a rectangle.