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A, B, C, D Are Mid-points of Sides of Parallelogram Pqrs. If Ar (Pqrs) = 36 Cm2, Then Ar (Abcd) = - Mathematics

MCQ

A, B, C, D are mid-points of sides of parallelogram PQRS. If ar (PQRS) = 36 cm2, then ar (ABCD) =

Options

  •  24 cm2

  •  18cm2

  •  30 cm2

  •  36 cm2

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Solution

Given:

(1) PQRS is a parallelogram.

(2) A, B, C, D are the midpoints of the adjacent sides of Parallelogram PQRS.

(3)  `ar ("||"^(gm) PQRS) = 36  cm^2` 

To find:  `AR("||"^(gm) ABCD)`

Calculation:

A and C are the midpoints of PS and QR respectively.

`AP = 1/2 SP`

`BP = 1/2 QR`

Now PQRS is a parallelogram which means

     `PS = QR `

`1/2PS = 1/2 QR`

AP = CQ ……..(1)

Also, PS || QR

AP || CQ ……(2)

From 1 and 2 we get that APCQ is a parallelogram.

Since Parallelogram APCQ and ΔABC are on the base AC and between the same parallels AC and PQ.

`∴ ar (ΔABC) = 1/2 ar ("||"^(gm) APCQ)`   ……(3)

Similarly ,

`ar  (ΔADC ) = 1/2 ar ("||"^(gm) ACRS)`   ……(4)

Adding 3 and 4 we get,

`ar (ΔABC ) + ar (ΔADC) = 1/2 ar ("||"^(gm) APCQ) + 1/2 ar("||"^(gm) ACRS)`

                         `ar (ABCD ) = 1/2 (ar ("||"^(gm) APCQ ) + ar ("||"^(gm) ACRS))`

                          `ar (ABCD ) = 1/2 (ar("||"^(gm) PQRS))`

`ar (ABCD ) = 1/2 (36)`

`ar (ABCD) = 18  cm^2`

 

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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 14 Areas of Parallelograms and Triangles
Q 8 | Page 61
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