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The dimensions of a rectangle ABCD are 51 cm × 25 cm. A trapezium PQCD with its parallel sides QC and PD in the ratio 9 : 8, is cut off from the rectangle as shown in the Fig. 12.6. If the area of the trapezium PQCD is 5/6 th part of the area of the rectangle, find the lengths QC and PD.

Solution:

Given, ABCD is a rectangle

The dimensions are 51 cm × 25 cm

A trapezium PQCD is cut off from the rectangle

The parallel sides of the trapezium QC and PD are in the ratio 9 : 8

The area of the trapezium PQCD is 5/6^th part of the area of the rectangle.

We have to find the lengths QC and PD.

The dimensions of a rectangle ABCD are 51 cm × 25 cm. A trapezium PQCD with its parallel sides QC and PD in the ratio 9 : 8, is cut off from the rectangle as shown in the Fig. 12.6. If the area of the trapezium PQCD is 5/6 th part of the area of the rectangle, find the lengths QC and PD.

Given, QC : PD = 9 : 8

Let the length of the parallel sides be

QC = 9x

PD = 8x

Area of rectangle = length × breadth

Area of rectangle ABCD = AD × AB

= 51 × 25

= 1275 cm²

Area of trapezium = 1/2 × sum of parallel sides × distance between parallel sides

Area of trapezium PQCD = 1/2 × (QC + PD) × DC

= 1/2 × (9x + 8x) × 25

= 1/2 × 17x × 25

Given, area of trapezium PQCD = 5/6 × area of rectangle ABCD

1/2 × 17x × 25 = 5/6 × 1275

1/2 × 17x = (5 × 1275)/(6 × 25)

1/2 × 17x = (5 × 51)/6

17x = (5 × 51 × 2)/6

17x = (5 × 51)/3

17x = 5 × 17

x = 5

So, QC = 9x = 9(5) = 45 cm

PD = 8x = 8(5) = 40 cm

Therefore, the lengths of QC and PD are 45 cm and 40 cm.

✦ Try This: Find the area of a trapezium ABCD in which AB∣∣DC, AB=77cm, BC=25cm, CD=60cm and DA=26cm.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 12

NCERT Exemplar Class 9 Maths Exercise 12.4 Problem 7

The dimensions of a rectangle ABCD are 51 cm × 25 cm. A trapezium PQCD with its parallel sides QC and PD in the ratio 9 : 8, is cut off from the rectangle as shown in the Fig. 12.6. If the area of the trapezium PQCD is 5/6 th part of the area of the rectangle, find the lengths QC and PD.

Summary:

The dimensions of a rectangle ABCD are 51 cm × 25 cm. A trapezium PQCD with its parallel sides QC and PD in the ratio 9 : 8, is cut off from the rectangle as shown in the Fig. 12.6. If the area of the trapezium PQCD is 5/6 th part of the area of the rectangle, the lengths QC and PD are 45 cm and 40 cm

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