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In Fig. 9.17, PQRS and ABRS are paralle...

In Fig. 9.17, `PQRS` and `ABRS` are parallelograms and X is any point on side BR. Show that (i) `a r(P Q R S) = a r(A B R S)`(ii) `a r(A XS)\ =1/2 ar(P Q R S)`

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If a triangle and a parallelogram are on the same base and between the same parallel lines, the area of the triangle will be half of that of a parallelogram or if two parallelograms are on the same and between two parallel lines then their area will be equal.
i) It can be observed from the figure that parallelogram `PQRS` and `ABRS` lie on the same base `SR` and also, they lie between the same parallel lines `SR` and `PB`.
According to Theorem 9.1, Parallelograms on the same base and between the same parallel lines are equal in area.
`Area (PQRS) = Area (ABRS)` .....(1)
ii) Consider `triangleAXS` and parallelogram `ABRS`
As both lie on the same base AS as well as between the same parallel lines `AS` and `BR`,
`Area (triangleAXS) = 1/2 Area (ABRS) .....(2)`
From Equations (1) and (2),
we obtain `Area (triangleAXS) = 1/2 Area (PQRS)`
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