Question

Prove it guys.......​

Answer 1

Step-by-step explanation:

the quadrilateral is square

so by joining midpoints of adjacent sides you will get a square of side square root of 2 times the side of the squarr

Answer 2

SOLUTION:-

Given:

A quadrilateral ABCD whose diagonals AC & BD are perpendicular to each other at O. P,Q,R & S are midpoints of side AB, BC,CD & DA respectively are joined are formed quadrilateral PQRS.

To prove:

PQRS is a rectangle.

Proof:

In ∆ABC

P & Q are mid-points of AB & BC respectively.

∴PQ||AC and PQ= 1/2AC.....(1)

[Mid-point theorem]

In ∆ACD,

R & S are mid-points of CD & DA respectively.

∴SR||AC and SR= 1/2AC.......(2)

[Mid-points theorem]

From (1) & (2), we have

PQ||SR & PQ= SR

Thus,

One pair of opposite sides of quadrilateral PQRS are parallel & equal.

∴PQRS is a parallelogram.

Since, PQ||AC= PM||NO

In ∆ABD,

P & S are mid-points of AB & AD respectively.

∴PS||BD [mid-points theorem]

=) PN||MO

∴Opposite sides of quadrilateral PMON parallel.

∴PMON is a parallelogram.

∴∠MPN = ∠MON

[Opposite angles of ||gm are equal]

But ∠MON= 90° [give]

∴∠MPN = 90°=∠QPS = 90°

Thus,

PQRS is a parallelogram whose one angle is 90°.

∴PQRS is a rectangle.

Hope it helps ☺️