# In Fig.8.3, X and Y are respectively the mid-points of the opposite sides AD and BC of a parallelogram ABCD. Also, BX and DY intersect AC at P and Q, respectively. Show that AP = PQ = QC.

**Solution:**

Given, ABCD is a __parallelogram__

X and Y are the midpoints of the opposite sides AD and BC of the parallelogram

BX and DY intersect AC at P and Q

We have to show that AP = PQ = QC

We know that opposite sides of a parallelogram are equal.

So, AD = BC ----------- (1)

Since X is the midpoint of AD

AX = DX

So, AD = AX + DX

AD = DX + DX

AD = 2DX --------------- (2)

Since Y is the midpoint of BC

BY = CY

So, BC = BY + CY

BC = BY + BY

BC = 2BY ------------------ (3)

Using (2) and (3) in (1),

2DX = 2BY

DX = BY

We know that a pair of opposite sides are equal and __parallel__ in a parallelogram.

So, DX || BY

This implies XBYD is a parallelogram

So, PX || QD

From triangle AQD,

Since X is the midpoint of AD

AP = PQ ------------ (4)

Similarly, from triangle CPB

CQ = PQ ------------ (5)

From (4) and (5),

AP = PQ = CQ

Therefore, AP = PQ = QC

**✦ Try This: **In ΔABC, AB = 5 cm, BC = 8 cm and CA = 7 cm. If D and E are respectively the mid-points of AB and BC, determine the length of DE.

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.3 Sample Problem 2**

## In Fig.8.3, X and Y are respectively the mid-points of the opposite sides AD and BC of a parallelogram ABCD. Also, BX and DY intersect AC at P and Q, respectively. Show that AP = PQ = QC.

**Summary:**

In Fig.8.3, X and Y are respectively the mid-points of the opposite sides AD and BC of a parallelogram ABCD. Also, BX and DY intersect AC at P and Q, respectively. It is shown that AP = PQ = QC

**☛ Related Questions:**

- In Fig.8.4, AX and CY are respectively the bisectors of the opposite angles A and C of a parallelogr . . . .
- One angle of a quadrilateral is of 108º and the remaining three angles are equal. Find each of the t . . . .
- ABCD is a trapezium in which AB || DC and ∠A = ∠B = 45º. Find angles C and D of the trapezium

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