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

Answer: (D)

$2:1$

Answer 2

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): $2:1$