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