*Q.1.In the given figure, PQRS is a parallelogram, PE⊥SRandRF⊥PS..If PQ = 16 cm, PE = 8 cm & RF = 10 cm. Calculate AD.*

**Solution:**

Area of the parallelogram

Now, area of parallelogram PQRS=

From equation (1) and (2), we get

*Q.2.If E, F, G and H are the mid-points respectively, of the sides of a parallelogram ABCD show that Area(EFGH)=12Area(ABCD)*

**Solution:**

Lets join HF.

In the parallelogram, i.e ABCD, AD = BC and AD || BC (because in a parallelogram the opposite sides are equal and parallel)

Since

Similarly, we can prove that,

Add Equation (1) and Equation (2), we obtain

*Q.3. DC and AD are two sides on which P and Q are two points lying respectively of a parallelogram ABCD. Prove that area(APB)=area(BQC)*

**Solution:**

It is observed that

Similarly,we can say that ∆APB and parallelogram ABCD lie between the same lines AB and DC that are parallel and on the same base AB .

Equating both the equations, i.e

Hence, proved.

*Q.4. In the given figure, intheinteriorofaparallelogramABCD,thereexistapointP.Show that*

*(i) area(APB)+area(PCD)=12area(ABCD)*

*(ii) area(APD)+area(PBC)=area(APB)+area(PCD)*

*[Hint: Draw a line i.e. parallel to AB, through P]*

Solution:

- A line segment EF is drawn, parallel to line segment AB and passing through point P.

In the parallelogram ABCD,

By equating , Equation (1) and Equation(2) , we get,

Therefore, quad ABFE is a parallelogram.

Similarly

Add equation(3) and equation(4), we get,

(ii)

A line segment MN is drawn

In the parallelogram ABCD,

By equating , Equation (6) and Equation(7) , we get,

Therefore

It can be said that ∆APD and parallelogram AMND are between

Similarly

Add equation(8) and equation(9), we get,

Now compare equation(5) with equation(10), we get

*Q.5. In the figure given below X is a point on the side BR and ABRS and PQRS are parallelograms. Prove that*

*(i) area (PQRS) = area (ABRS)*

*(ii) area(ΔPXS)=12area(PQRS)*

**Solution:**

(i)It can be said that

(ii)Consider

As these lie on the same base and are between the same parallel lines AS and BR

By equating , equation (1) and equation(2)

*Q.6. A farmer had a field and that was in a parallelogram shape PQRS .He took a point A on RS and he joined it to points P and Q. In how many parts the field is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately.How should he do it ?*

**Solution:**

From the figure

The parts which are triangular in shape are –

We know that if a parallelogram and a triangle are between the same parallels and on the same base

By equation , equation (i) and (ii)

Clearly