Question

In the given figure LMNO and PMNQ are two parallelograms and R is any point on MP. Show that area of triangle NRQ is equal to half the area of parallelogram LMNO.

Answer 1

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Answer 2

If LMNO and PMNQ are two parallelograms and R is any point on MP, then the area of ΔNRQ = 1/2 * area of parallelogram LMNO.

Step-by-step explanation:

Step 1: Proving ar(LMNO) = ar(PMNQ)

From the given figure, we can say that

Both the parallelogram LMNO and PMNQ are on the same base MN and lie between the same parallel lines MN and LQ.

We know that if two parallelograms are on the same base and lie between the same parallel lines, then they have the same area.

∴ Area (parallelogram LMNO) = Area (parallelogram PMNQ) …. (i)

Step 2: Proving ar(NRQ) = 1/2 ar(LMNO)

From the given figure, we can say that  

The ΔNRQ and the parallelogram PMNQ have the same base NQ and lie between the same parallel lines PM and NQ.

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

∴ Area (∆ NRQ) = ½ * Area (parallelogram PMNQ) …… (ii)

Comparing eq. (i) & (ii), we get

Area (∆ NRQ) = ½ * Area (parallelogram LMNO)

Hence proved

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