Advertisements
Advertisements
प्रश्न
PQRS is a parallelogram. T is the mid-point of RS and M is a point on the diagonal PR such that MR = `(1)/(4)"PR"`. TM is joined and extended to cut QR at N. Prove that QN = RN.
Advertisements
उत्तर
Join PR to intersect QS at O
Diagonals of a parallelogram bisect each other.
Therefore, OP = OR
But MR = `(1)/(4)"PR"`
∴ MR = `(1)/(4)(2 xx "QR")`
⇒ MR = `(1)/(2)"OR"`
Hence, M is the mid-point of OR.
In ΔROS, T and M are the mid-points of RS and OR respectively.
Therefore, TM || OS
⇒ TN || QS
Also in ΔRQS, T is the mid-point of RS and TN || QS
Therefore, N is the mid-point of QR and TN = `(1)/(2)"QS"`
⇒ QN = RN.
APPEARS IN
संबंधित प्रश्न
ABCD is a rectangle with ∠ADB = 55°, calculate ∠ABD.
ABCD is a quadrilateral P, Q, R and S are the mid-points of AB, BC, CD and AD. Prove that PQRS is a parallelogram.
P is a point on side KN of a parallelogram KLMN such that KP : PN is 1 : 2. Q is a point on side LM such that LQ : MQ is 2 : 1. Prove that KQMP is a parallelogram.
In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that
MN bisects QS.
In the given figure, PQRS is a parallelogram in which PA = AB = Prove that: SA ‖ QB and SA = QB.
In the given figure, PQRS is a parallelogram in which PA = AB = Prove that: SAQB is a parallelogram.
Prove that the diagonals of a square are equal and perpendicular to each other.
The diagonals AC and BC of a quadrilateral ABCD intersect at O. Prove that if BO = OD, then areas of ΔABC an ΔADC area equal.
In ΔABC, the mid-points of AB, BC and AC are P, Q and R respectively. Prove that BQRP is a parallelogram and that its area is half of ΔABC.
The medians QM and RN of ΔPQR intersect at O. Prove that: area of ΔROQ = area of quadrilateral PMON.
