A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is?
Explanation:
Let the longer side of the rectangle is 2a and the shorter side is b as shown in the figure.
Using pythagoras theorem, we get a2 + b2 = 22 = 4 ...(1)
Also, AM ≥ GM ⇒ a2+b22 ≥ a2b2
⇒ a2 + b2 ≥ 2ab
⇒ 4 ≥ 2ab
⇒ 2 ≥ ab
∴ Highest value of ab = 2, when a = b.
⇒ Ratio of longer side to shorter side = 2a : b = 2 : 1.
Hence, option (a).
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