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 length of the rectangle be \( l \) and breadth be \( b \).
The radius, \( \frac{l}{2} \), and \( b \) in the above diagram form a right-angled triangle. 
Using the Pythagorean theorem:
\[ \left(\frac{l}{2}\right)^2 + b^2 = 2^2 \] 
Area of the rectangle = \( l \times b \)

The area can be maximized by using the AM-GM (Arithmetic Mean – Geometric Mean) inequality or by setting the two squares equal to make their geometric mean maximum:
\[ \left(\frac{l}{2}\right)^2 = b^2 \] 
Solving this gives:
\[ \frac{l}{2} = b \Rightarrow l = 2b \] 
So, the ratio of length to breadth is:
\[ \frac{l}{b} = \frac{2}{1} \]

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