In the figure below, ABCD is a parallelogram and X, Y are the midpoints of AB, CD. The lines DX and BY cut AC at P and Q.

Prove that AP = PQ = QC.

Open in App

Solution

Given: ABCD is a parallelogram with X and Y as the midpoints of AB and CD.

∴ AX = XB and CY = YD

Also, AB = DC (Opposite sides of a parallelogram are equal)

⇒ AX + XB = CY + YD

2XB = 2DY

XB = DY

In quadrilateral XBYD, XB = DY

XB || DY (As AB || DC)

We know that if a pair of opposite sides are equal and parallel then the quadrilateral is a parallelogram.

∴ XBYD is a parallelogram.

⇒ DX || YB

In ΔABQ, X is the midpoint of AB and XP || BQ. (As DX || YB)

We know that in a triangle, a line through the midpoint of a side and parallel to another side bisects the third side.

∴ AP = PQ …(1)

In ΔCDP, Y is the midpoint of CD and YQ || DP.

∴ CQ = QP …(2)

From equation (1) and (2), we have:

AP = PQ = QC

Suggest Corrections

Similar questions

P. Q, R and S are respectively the midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD. Show that

(i) PQ || AC and $P Q = \frac{1}{2} A C$

(ii) PQ || SR

(iii) PQRS is a parallelogram.

In the adjoining figure, $A B C D$ is a ||gm in which $E$ and $F$ are the midpoints of $A B$ and $C D$ respectively. If $G H$ is a line segment that cuts $A D, E F$ and $B C$ at $G, P$ and $H$ respectively, prove that $G P = P H$ .

Q. ABCD is a parallelogram . AB is divided at P and CD at Q so that AP : PB = 3 : 2 and CQ : QD = 4 : 1 . If PQ meets AC at R then prove that AR =

\frac{3}{7}

AC .

Q. In the adjoining figure, ABCD is a || gm in which E and F are the midpoints of AB and CD respectively. If GH is a line segment that cuts AD, EF and BC at G, P and H respectively, prove that GP = PH.

Q. Points

P

and

Q

have been taken on opposite sides

A B

and

C D

, respectively of a parallelogram

A B C D

such that

A P = C Q

. Show that

A C

and

P Q

disect each other.

Join BYJU'S Learning Program

In the figure below, ABCD is a parallelogram and X, Y are the midpoints of AB, CD. The lines DX and BY cut AC at P and Q. Prove that AP = PQ = QC.

In the figure below, ABCD is a parallelogram and X, Y are the midpoints of AB, CD. The lines DX and BY cut AC at P and Q.

Prove that AP = PQ = QC.