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A quadrilateral A B C D is such that dia...

A quadrilateral `A B C D` is such that diagonal `B D` divides its area in two equal parts. Prove that `B D` bisects `A Cdot` GIVEN : A quadrilateral `A B C D` in which diagonal `B D` bisects it. i.e. `a r( A B D)=a r( B D C)` CONSTRUCTION : Join `A C` Suppose `A C` and `B D` intersect at `O` . Draw `A L_|_B D` and `C M B Ddot` TO PROVE : `A O=O Cdot`

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Given: `BD` is the diagonal of quadrilateral `ABCD`, which divide it into two equal areas.From `A` and `C` draw perpendiculars to the diagonal `BD`.
`AOE=COF` (Vertical opposite angles)
In `triangleAOE` and `triangleCOF`,
`angleAEO=angleCFO` (Both are `90^@`)
`AE=CF` (Bases and areas of both the `triangleADB` and `triangleBDC` are same.)
`EAO=FCO` `(angleAEO=angleCFO=90°` and `angleAOE=angleCOF)`
`trinagleAOEcongtriangleCOF ` ( By `ASA`)
`AO=OC` (by `CPCT`)
Thus, `BD` bisect `AC` at `O`.
Hence Proved.
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