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Let ABC be a triangle of area 24 sq. units and PQR be the triangle formed by the mid-points of the sides of Δ ABC. Then the area of ΔPQR is - Mathematics

MCQ

Let ABC be a triangle of area 24 sq. units and PQR be the triangle formed by the mid-points of the sides of Δ ABC. Then the area of ΔPQR is

Options

  • 12 sq. units

  •  6 sq. units

  • 4 sq. units

  • 3 sq. units

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Solution

Given: (1) The Area of ΔABC = 24 sq units.

(2) ΔPQR is formed by joining the midpoints of ΔABC

To find: The area of ΔPQR

Calculation: In ΔABC, we have

Since Q and R are the midpoints of BC and AC respectively.

∴  PQ || BA ⇒  PQ || BP

Similarly, RQ || BP. So BQRP is a parallelogram.

Similarly APRQ and PQCR are parallelograms.

We know that diagonal of a parallelogram bisect the parallelogram into two triangles of equal area.

Now, PR is a diagonal of ||gmAPQR.

∴ Area of ΔAPR = Area of ΔPQR ……(1)

Similarly,

PQ is a diagonal of ||gm PBQR

∴ Area of ΔPQR = Area of ΔPBQ ……(2)

QR is the diagonal of ||gm  PQCR

∴ Area of ΔPQR = Area of ΔRCQ ……(3)

From (1), (2), (3) we have

Area of ΔAPR = Area of ΔPQR = Area of ΔPBQ = Area of ΔRCQ

But

Area of ΔAPR + Area of ΔPQR + Area of ΔPBQ + Area of ΔRCQ = Area of ΔABC

4(Area of ΔPBQ) = Area of ΔABC

`= 1/4 `Area of ΔABC 

 

∴ Area of ΔPBQ  `= 1/4 (24)`

= 6 sq units

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 14 Areas of Parallelograms and Triangles
Q 3 | Page 60
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