6. The diagonals of a quadrilateral intersect at
right angles. Prove that the figure obtained by
joining the mid-points of the adjacent sides of
the quadrilateral is a rectangle.
Answers
ANSWER:-
Given:
The diagonals of a quadrilateral intersected at right angles.
To prove:
Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.
Proof:
In ∆ABC,
P & Q are midpoints of AB & BC respectively.
Therefore,
PQ||AC
PQ= 1/2AC........(1) [Mid-point theorem]
&
In ∆ACD,
R & S are mid-points of CD & DA respectively.
Therefore,
SR||AC &
SR= 1/2AC.......(2) [mid-points Theorem]
From (1) & (2), we get;
PQ||SR & PQ= SR
Thus,
One pair of opposite sides of quadrilateral PQRS are parallel & equal.
Therefore,
PQRS is a parallelogram.
Since, PQ||AC &
PM||NO
In ∆ABD
P& S are mid-points of AB & AD respectively.
Therefore,
PS||MO [mid-points Theorem]
=) PN||MO
Therefore,
Opposite sides of quadrilateral PMON parallel.
Therefore,
PMON is a parallelogram.
Therefore,
MPN = MON
[opposite angles of ||gm are equal]
But,
MON =90° [give]
MPN = 90°
QPS = 90°
Thus,
PQRS is a parallelogram whose one angles is 90°.
Therefore,
PQRS is a rectangle.
Proved!
Hope it helps ☺️
Step-by-step explanation:
follow this simple and easy problem steps
