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State true or false

The diagonals of a quadrilateral intersect at right angles. the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.

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True

False

answer is A.

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Detailed Solution

Given,
A quadrilateral = ABCD
Diagonals = AC, BD
Mid points of quadrilateral = P, Q, R, and S.
Question Image

In the triangle

△ ABC and △ ADC

From mid-point theorem

2 PQ = AC and PQ ∥ AC

…………. (1)

2 RS = AC and RS ∥ AC

…………… (2)
Form the equation (1) and (2)

PQ = RS and PQ ∥ RS

Also, similarly we have

PS = RQ and PS ∥ RQ

PQRS is a parallelogram. So,

PQ ∥ AC

∠ AOD = ∠ PXO = 90^{\circ}

Again,

BD ∥ RQ

∠ PXO = ∠ RQX = 90^{\circ}

So, we get,

∠ QRS = ∠ RSP = ∠ SPQ = 90^{\circ}

Here, each angle is equal to 90 degree. So, we can say that PQRS is a rectangle.
Hence the given statement is true.
Correct option is 1.

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