ABCD is a quadrilateral whose diagonals AC and BD intersect at O. If triangles AOB and COD have areas 4 and 9 respectively, then the minimum area that ABCD can have is
Explanation:
Area of triangle AOB = 4, DOC = 9, AOD = x and BOC = y
In a quadrilateral, product of area of diagonally opposite triangles is same.
∴ 4 × 9 = x × y ⇒ xy = 36
Area of quadrilateral = 4 + 9 + x + y Now we need to minimise 4 + 9 + x + y This is possible when x + y is least possible.
We know, AM ≥ GM ⇒ (x + y)2 ≥ xy ⇒ (x + y) ≥ 12 ∴ Least possible value of x + y = 12
⇒ Least possible area of the quadrilateral = 4 + 9 + x + y = 25
Hence, option (b).
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