Advertisements
Advertisements
Question
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus.
Advertisements
Solution
Given: In a quadrilateral ABCD, P, Q, R and S are the mid-points of sides AB, BC, CD and DA, respectively.
Also, AC = BD
To prove: PQRS is a rhombus.
Proof: In ΔADC, S and R are the mid-points of AD and DC respectively.
Then, by mid-point theorem.
SR || AC and SR = `1/2` AC ...(i)
In ΔABC, P and Q are the mid-points of AB and BC respectively.
Then, by mid-point theorem.
PQ || AC and PQ = `1/2` AC ...(ii)
From equations (i) and (ii),
SR = PQ = `1/2` AC ...(iii)
Similarly, in ΔBCD,
RQ || BD and RQ = `1/2` BD ...(iv)
And in ΔBAD,
SP || BD and SP = `1/2` BD ...(v)
From equations (iv) and (v),
SP = RQ = `1/2` BD = `1/2` AC [Given, AC = BD] ...(vi)
From equations (iii) and (vi),
SR = PQ = SP = RQ
It shows that all sides of a quadrilateral PQRS are equal.
Hence, PQRS is a rhombus.
APPEARS IN
RELATED QUESTIONS
ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that:
- SR || AC and SR = `1/2AC`
- PQ = SR
- PQRS is a parallelogram.
In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.
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 a ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing
through A. If L is the mid-point of BC, prove that ML = NL.
The side AC of a triangle ABC is produced to point E so that CE = AC. D is the mid-point of BC and ED produced meets AB at F. Lines through D and C are drawn parallel to AB which meet AC at point P and EF at point R respectively.
Prove that:
- 3DF = EF
- 4CR = AB
In triangle ABC; M is mid-point of AB, N is mid-point of AC and D is any point in base BC. Use the intercept Theorem to show that MN bisects AD.
ABCD is a parallelogram.E is the mid-point of CD and P is a point on AC such that PC = `(1)/(4)"AC"`. EP produced meets BC at F. Prove that: 2EF = BD.
ABCD is a kite in which BC = CD, AB = AD. E, F and G are the mid-points of CD, BC and AB respectively. Prove that: ∠EFG = 90°
In the given figure, T is the midpoint of QR. Side PR of ΔPQR is extended to S such that R divides PS in the ratio 2:1. TV and WR are drawn parallel to PQ. Prove that T divides SU in the ratio 2:1 and WR = `(1)/(4)"PQ"`.
Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.
