Question

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

• A. \(1 :1\)
• B. \(\sqrt5 :1\)
• C. \(\sqrt2 :1\)
• D. \(2:1\)

Answer 1

The Correct Option is D

Let the lenght of the rectangle be \(l\) and breadth be \(b\).
The radius, \(\frac l2\) and \(b\) in the above diagram form a right-angled triangle.
\((\frac l2)^2 + b^2 = 2^2\)
Area of the rectangle \(= lb\)
Area of the rectangleh can be obtained by considering 2 times the geometric mean of \((\frac l2)^2\) and \(b^2\).
So, for the maximum area,
\((\frac l2)^2=b^2\)

\(⇒\frac l2=b\)
\(⇒l=2b\)
\(⇒\frac lb = \frac 21\)

So, the correct option is (D): \(2:1\)