Question

show that if the diagonals of a quadrilateral PQRS is a rectangle​

Answer 1

Answer:

i think the question is incomplete

Answer 2

$\huge\underline\bold \red{\star AnsweR \star }$

Let us consider a quadrilateral ABCD whose diagonal are perpendicular to each other AC & BD.

$\large\bf \pink{To \: prove :}$ PQRS is a rectangle.

Proof :- In ∆ ABC , P and Q are mid points of AB & BC respectively.

$\therefore$ PQ || AC and PQ = $\large\frac {1 }{2 AC }$ ------ (i)$\bf\pink{[ Mid \: point \: theorem ]}$

Further in ∆ ACD , R and S are mid points CD and DA respectively.

$\therefore$ SR || AC and SR $\large\frac {1 }{2 AC }$ ------ (ii) $\bf\pink{[ Mid \: point \: theorem ]}$

From (i) and (ii) we have PQ || SR & PQ = SR .

Thus, one pair of opposite sides of quadrilateral PQRS are parallel and equal.

$\therefore$ PQRS is a parallelogram.

Since, PQ || AC $\implies$ PM || NO

In ∆ ABD , P and S are mid point of AB and AD respectively.

$\therefore$ PS || BD [mid point theorem]

$\implies$ PN || MO

$\therefore$ Opposite sides of quadrilateral PMON are parallel.

$\therefore$ PMON is a parallelogram.

$\therefore \sf \angle MON = \angle MPN \bf\pink{[Opposite \: sides \: of \: parallelogram \: are \:equal.]}$

But , $\bf\angle MON = 90 \degree$

Thus, PQRS is a Parallelogram whose angle is 90°

$\large\bf\blue{ \therefore PQRS \: is \: a \: rectangle}$$\large\blue {✓}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf{\underline{\underline \pink {Hence \: proved ....}}}$

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Hope this helps you $\red {✓}$

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