Advertisements
Advertisements
Question
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 rectangle.
Advertisements
Solution
The figure is shown below
Let ABCD be a quadrilateral where P, Q, R, S are the midpoint of AB, BC, CD, DA. Diagonal AC and BD intersect at a right angle at point O. We need to show that PQRS is a rectangle
Proof:
From and ΔABC and ΔADC
2PQ = AC and PQ || AC …..(1)
2RS = AC and RS || AC …..(2)
From (1) and (2) we get,
PQ = RS and PQ || RS
Similarly, we can show that PS=RQ and PS || RQ
Therefore PQRS is a parallelogram.
Now PQ || AC, therefore ∠AOD = ∠PXO = 90° ...[ Corresponding angel ]
Again BD || RQ, therefore ∠PXO = ∠RQX = 90° ...[ Corresponding angel]
Similarly ∠QRS = ∠RSP = ∠SPQ = 90°
Therefore PQRS is a rectangle.
Hence proved.
APPEARS IN
RELATED QUESTIONS
In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC =
21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadrilateral ARPQ.
In Fig. below, BE ⊥ AC. AD is any line from A to BC intersecting BE in H. P, Q and R are
respectively the mid-points of AH, AB and BC. Prove that ∠PQR = 90°.
In the below Fig, ABCD and PQRC are rectangles and Q is the mid-point of Prove thaT
i) DP = PC (ii) PR = `1/2` AC
In triangle ABC, M is mid-point of AB and a straight line through M and parallel to BC cuts AC in N. Find the lengths of AN and MN if Bc = 7 cm and Ac = 5 cm.
In parallelogram ABCD, E and F are mid-points of the sides AB and CD respectively. The line segments AF and BF meet the line segments ED and EC at points G and H respectively.
Prove that:
(i) Triangles HEB and FHC are congruent;
(ii) GEHF is a parallelogram.
The diagonals of a quadrilateral intersect each other at right angle. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.
Show that the quadrilateral formed by joining the mid-points of the adjacent sides of a square is also a square.
In the given figure, PS = 3RS. M is the midpoint of QR. If TR || MN || QP, then prove that:
ST = `(1)/(3)"LS"`
P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC = BD and AC ⊥ BD. Prove that PQRS is a square.
P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.
