Question

The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a square only if, diagonals of ABCD are ___________ and _________.

Answer 1

Given:
ABCD is a quadrilateral
The quadrilateral formed by joining the mid points of the sides of a quadrilateral ABCD, is a square.
Let the square be PQRS.



P is the mid-point of AB, Q is the mid-point of BC, R is the mid-point of CD and S is the mid-point of AD.

Using mid-point theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is equal to the half of it.

In ∆ABC,
PQ = $\frac{1}{2}$AC

and In ∆BCD,
QR = $\frac{1}{2}$BD

Since, PQ = QR (sides of a square are equal)
Therefore, $\frac{1}{2}$AC = $\frac{1}{2}$BD
⇒ AC = BC

Hence, the diagonals are equal.
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In ∆ABC,
PQ || AC

and In ∆BCD,
QR || BD

Since, PQ$\perp$QR
Therefore, AC$\perp$BD

Hence, the diagonals are perpendicular.

Hence, the figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a square only if, diagonals of ABCD are perpendicular and equal.