Advertisements
Advertisements
Question
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.
Advertisements
Solution
Join AC and BD
In ΔABC,
P and Q are mid-point of AB and BC respectively.
Therefore, PQ || AC and PQ = `(1)/(2)"AC"` ........(i)
In ΔADC,
S and R are mid-point of AD and DC respectively.
Therefore, SR || AC and SR = `(1)/(2)"AC"` ........(ii)
From (i) and (ii)
PQ || SR and PQ = SR
Therefore, PQRS is a parallelogram.
APPEARS IN
RELATED QUESTIONS
Prove that if the diagonals of a parallelogram are equal then it is a rectangle.
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.
Prove that the line segment joining the mid-points of the diagonals of a trapezium is parallel to each of the parallel sides, and is equal to half the difference of these sides.
In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that
RN and RM trisect QS.
ABCD is a trapezium in which side AB is parallel to side DC. P is the mid-point of side AD. IF Q is a point on the Side BC such that the segment PQ is parallel to DC, prove that PQ = `(1)/(2)("AB" + "DC")`.
In the given figure, PQRS is a parallelogram in which PA = AB = Prove that: SAQB is a parallelogram.
Prove that the diagonals of a kite intersect each other at right angles.
In the given figure, AB ∥ SQ ∥ DC and AD ∥ PR ∥ BC. If the area of quadrilateral ABCD is 24 square units, find the area of quadrilateral PQRS.
In the given figure, PQ ∥ SR ∥ MN, PS ∥ QM and SM ∥ PN. Prove that: ar. (SMNT) = ar. (PQRS).
In ΔPQR, PS is a median. T is the mid-point of SR and M is the mid-point of PT. Prove that: ΔPMR = `(1)/(8)Δ"PQR"`.
