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Question
Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is `1/2 |bar"AC" xx bar"BD"|`
Solution
Vector area of quadrilateral ABCD = {Vector area of ∆ABC} + {Vector area of ∆ACD}
= `1/2 (bar"AB" xx bar"AC") + 1/2(bar"AC" xx bar"AD")`
= `- 1/2 (bar"AC" xx bar"AB") + 1/2(bar"AC" xx bar"AD")`
= `1/2bar"AC" xx [- bar"AB" + bar"AD"]`
= `1/2bar"AC" xx [bar"BA" + bar"AD"]`
= `1/2 bar"AC" xx bar"BD"`
∴ The area of the quadrilateral ABCD = `1/2|bar"AC" xx bar"BD"|`
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