Q. 64.4( 279 Votes )

# Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Answer :

**Mid point theorem:**The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the third side.

Let ABCD is a quadrilateral in which P, Q, R, and S are the mid-points of sides AB, BC, CD, and DA respectively. Join PQ, QR, RS, SP, and BD

In ΔABD, S and P are the mid-points of AD and AB respectively. Therefore, by using mid-point theorem:

SP || BD and SP = BD (1)

Similarly in ΔBCD,

QR || BD and QR = BD (2)

From equations (1) and (2), we obtain

SP || QR and SP = QR

In quadrilateral SPQR, one pair of opposite sides is equal and parallel to each other

Therefore, SPQR is a parallelogram.

We know that diagonals of a parallelogram bisect each other

Hence, PR and QS bisect each other.

Rate this question :

ABCD is a trapezium in which AB||DC. M and N are the mid-points of AD and BC respectively, If AB=12cm, MN=14 cm, then CD=

RD Sharma - MathematicsE and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Prove that EF || AB and EF = (AB + CD).

[Hint: Join BE and produce it to meet CD produced at G.]

NCERT Mathematics ExemplarIn 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.

RD Sharma - MathematicsIn Fig. 14.99, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ= AC. If PQ produced meets BC at R, Prove that R is a mid-point of BC.

RD Sharma - Mathematics

In Fig. 14.100, ABCD and PQRC are rectangles and Q is the mid-point of AC. Prove that

(i) DP = PC (ii) PR = AC.

RD Sharma - MathematicsProve that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

RS Aggarwal & V Aggarwal - MathematicsShow that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.

RS Aggarwal & V Aggarwal - MathematicsIn ΔABC, E is the mid-point of median AD such that BE produced meets AC at F. If AC = 10.5 cm, then AF=

RD Sharma - MathematicsIn the adjoining figure, are the midpoints of the sides and respectively, of Show that and

RS Aggarwal & V Aggarwal - Mathematics