Advertisements
Advertisements
Question
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.
Advertisements
Solution
Given: In a parallelogram ABCD, P and Q are the mid-points of AS and CD, respectively.
To show: PRQS is a parallelogram.
Proof: Since, ABCD is a parallelogram.
AB || CD
⇒ AP || QC
Also, AB = DC
`1/2`AB = `1/2`DC ...[Dividing both sides by 2]
⇒ AP = QC ...[Since, P and Q are the mid-points of AB and DC]
Now, AP || QC and AP = QC
Thus, APCQ is a parallelogram.
∴ AQ || PC or SQ || PR ...(i)
Again, AB || DC or BP || DQ
Also, AB = DC
⇒ `1/2`AB = `1/2`DC ...[Dividing both sides by 2]
⇒ BP = QD ...[Since, P and Q are the mid-points of AB and DC]
Now, BP || QD and BP = QD
So, BPDQ is a parallelogram.
∴ PD || BQ or PS || QR ...(ii)
From equations (i) and (ii),
SQ || RP and PS || QR
So, PRQS is a parallelogram.
Hence proved.
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 below fig. ABCD is a parallelogram and E is the mid-point of side B If DE and AB when produced meet at F, prove that AF = 2AB.
In Fig. below, triangle ABC is right-angled at B. Given that AB = 9 cm, AC = 15 cm and D,
E are the mid-points of the sides AB and AC respectively, calculate
(i) The length of BC (ii) The area of ΔADE.
D, E, and F are the mid-points of the sides AB, BC and CA of an isosceles ΔABC in which AB = BC.
Prove that ΔDEF is also isosceles.
ABCD is a quadrilateral in which AD = BC. E, F, G and H are the mid-points of AB, BD, CD and Ac respectively. Prove that EFGH is a rhombus.
In trapezium ABCD, sides AB and DC are parallel to each other. E is mid-point of AD and F is mid-point of BC.
Prove that: AB + DC = 2EF.
In parallelogram PQRS, L is mid-point of side SR and SN is drawn parallel to LQ which meets RQ produced at N and cuts side PQ at M. Prove that M is the mid-point of PQ.
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.
In the given figure, ABCD is a trapezium. P and Q are the midpoints of non-parallel side AD and BC respectively. Find: PQ, if AB = 12 cm and DC = 10 cm.
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. Prove that PQRS is a rectangle.
