A, B, C, D are mid-points of sides of parallelogram *PQRS*. If ar (*PQRS*) = 36 cm^{2}, then ar (*ABCD*) =

#### Options

24 cm

^{2}18cm

^{2}30 cm

^{2}36 cm

^{2}

#### 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`