Question

The area of the largest square that can be inscribed in a circle of radius 5 cm is

Answer 1

Answer:

Consider a semi-circle with a rectangle ABCD inscribed in it.

Let, O = centre of the semi-circle

A and B lies on the base of the semi-circle

OA = OB = x

D and C lie on the semi-circle

BC = AD = y

AB = CD = 2x

By Pythagorean theorem,

CB2 + OB2 = OC2

⇒ y2 + x2 = (5)2

⇒ y2 = 25 - x2

⇒ y = √(25 - x2)

Now, area of rectangle in terms of x,

Area, A = 2x × y

= 2x × √(25 - x2)

Differentiating,

A' = 2 × √(25 - x2) - 2x2/(25 - x2)

When x = 0, y = 5 and when x = 5, y = 0, area = 0.

It implies that area is maximum when the value of x lies between 0 and 5.

This will occur where A’ = 0.

⇒ 2 × √(25 - x2) - 2x2/(25 - x2) = 0

⇒ 2 × √(25 - x2) = 2x2/(25 - x2)

On simplification, we get,

⇒ 2 × (25 - x2) = 2x2

⇒ 25 - x2 = x2

⇒ 2x2 = 25

⇒ x2 = 25/2

⇒ x = √25/2

Now, y = √(25 - x2) becomes

⇒ y = √(25 - (√25/2)2)

⇒ y = √(25 - 25/2)

⇒ y = √(50 - 25)/2

⇒ y = √25/2

Maximum area = 2xy

= 2(√25/2)(√25/2)

= 25

Therefore, the area of the largest rectangle = 25 square units.

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