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Prove that the area of a rhombus is equal to half the rectangle contained by its diagonals.

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#### Solution

Since the diagonals of a rhombus intersect at right angles,

Therefore, OB ⊥ AC and OD ⊥AC

Now, ar(rhombus ABCD)

= ar(ΔABC) + ar(ΔADC)

= `(1)/(2)("AC" xx "BO") + (1)/(2)("AC" xx "DO")`

= `(1)/(2){"AC" xx ("BO" + "DO")}`

= `(1)/(2)("AC" xx "BD")`

Therefore, the area of a rhombus is equal to half the rectangle contained by its diagonals.

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