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In Fig. 9.20, ABCD is a parallelogram. Points P and Q on BC trisects BC in three equal parts. Prove that ar (APQ) = ar (DPQ) = 1/6 ar(ABCD)

In Fig. 9.20, ABCD is a parallelogram. Points P and Q on BC trisects BC in three equal parts. Prove that ar (APQ) = ar (DPQ) = 1/6 ar(ABCD)

Solution:

Given, ABCD is a parallelogram.

P and Q on BC trisects BC in three equal parts.

We have to prove that ar(APQ) = ar(DPQ) = 1/6 ar(ABCD)

Draw PR and QS parallel to AB, through P and Q

So, PQRS is a parallelogram

Given, P and Q trisects BC

So, PQ = 1/3 BC.

We know that, if a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half of the area of the parallelogram.

The triangle APD and parallelogram ABCD are on the same base BC and between the same parallels BC and AD

So, ar (APD) = 1/2 ar (ABCD) ------------ (1)

similarly,

ar (AQD) = 1/2 ar (ABCD) ----------------- (2)

From (1) and (2),

ar (APD) = ar (AQD) ------------------------- (3)

Subtracting ar (AOD) from both sides in (3),

ar (APD) - ar (AOD) = ar (AQD) - ar (AOD)

From the figure,

ar (APD) - ar(AOD) = ar (APO)

ar(AQD) - ar(AOD) = ar(OQD)

So, ar (APO) = ar (OQD) ------------------------ (4)

Adding ar (OPQ) on both sides in (4),

ar (APO) + ar (OPQ) = ar (OQD) + ar (OPQ)

From the figure,

ar(APO) + ar(OPQ) = ar(APQ)

ar(OQD) + ar(OPQ) = ar(DPQ)

ar (APQ) = ar (DPQ) ------------------------------ (5)

Since, ar (APQ) = 1/2 ar (PQRS),

From (5),

ar (DPQ) = 1/2 ar (PQRS)

Now, ar (PQRS) = 1/3 ar (ABCD)

So, ar (APQ) = ar (DPQ)

= 1/2 ar (PQRS)

= 1/2 × 1/3 ar (ABCD)

= 1/6 ar (ABCD)

Therefore, ar(APQ) = ar(DPQ) = 1/6 ar(ABCD)

✦ Try This: In Fig, D, E and F are the mid-points of the sides BC, CA and AB respectively of △ABC. If AB = 6.2 cm, BC = 5.6 cm and CA = 4.6 cm, find the perimeter of triangle DEF.

In Fig, D, E and F are the mid-points of the sides BC, CA and AB respectively of △ABC. If AB = 6.2 cm, BC = 5.6 cm and CA = 4.6 cm, find the perimeter of triangle DEF.

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

NCERT Exemplar Class 9 Maths Exercise 9.4 Sample Problem 1

In Fig. 9.20, ABCD is a parallelogram. Points P and Q on BC trisects BC in three equal parts. Prove that ar (APQ) = ar (DPQ) = 1/6 ar(ABCD)

Summary:

A parallelogram is a two-dimensional geometrical shape, whose sides are parallel to each other and equal in length. In Fig. 9.20, ABCD is a parallelogram. Points P and Q on BC trisects BC in three equal parts. It is proven that ar (APQ) = ar (DPQ) = 1/6 ar(ABCD)

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