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The vertices of a rectangle PQRS are joi...

The vertices of a rectangle PQRS are joined from an interior point 'O'. Prove that the sum of the area of two opposite triangles so formed is equal to the sum of the areas of remaining two triangles

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Through O, draw `MN bot PQ`
`:.` area of `Delta OPQ + " area of " Delta ORS`
`= (1)/(2) xx PQ xx OM + (1)/(2) xx RS xx ON`
`= (1)/(2) xx PQ xx OM + (1)/(2) xx PQ xx ON`
(`:. RS = PQ`, opposite sides of a rectangle)
`= (1)/(2) xx PQ xx (OM + ON) = (1)/(2) xx PQ xx MN`
`= (1)/(2) xx PQ xx QR = (1)/(2) xx` area of rectangle PQRS ....(1)
Again
area of `Delta OPS + " area of " Delta OQR = " area of " square PQRS - ( " area of " Delta OPQ + " area of " Delta ORS)`
= area of `square PQRS - (1)/(2) xx` area of `square PQRS` [from (1)]
`= (1)/(2) xx " area of " square PQRS` ...(2)
From (1) and (2), we get.
area of `Delta OPQ + " area of " Delta ORS = " area of " Delta OPS + " area of " Delta OQR`
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