Question

Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

• A. 24,10
• B. 25,9
• C. 24,12
• D. 25,10

Answer 1

The Correct Option is A

To determine the possible length and breadth of rectangle ABCD inscribed in a circle with radius 13 cm, we need to understand that the diagonal of the rectangle is equal to the diameter of the circle.

The formula for the diagonal \(d\) of a rectangle with length \(l\) and breadth \(b\) is given by:

$$d = \sqrt{l^2 + b^2}$$

Since the rectangle is inscribed in the circle, the diagonal is also the diameter of the circle:

$$d = 2 \times 13 = 26$$

Substituting \(d\) in the diagonal formula, we have:

$$\sqrt{l^2 + b^2} = 26$$

Squaring both sides, we get:

$$l^2 + b^2 = 676$$

We need to check which pair among the options satisfies this equation. Let's verify:

Option 1: 24, 10

• Calculate \(l^2 + b^2 = 24^2 + 10^2 = 576 + 100 = 676\)
• This satisfies the equation \(676 = 676\).

Option 2: 25, 9

• Calculate \(l^2 + b^2 = 25^2 + 9^2 = 625 + 81 = 706\)
• This does not satisfy the equation \(676 \neq 706\).

Option 3: 24, 12

• Calculate \(l^2 + b^2 = 24^2 + 12^2 = 576 + 144 = 720\)
• This does not satisfy the equation \(676 \neq 720\).

Option 4: 25, 10

• Calculate \(l^2 + b^2 = 25^2 + 10^2 = 625 + 100 = 725\)
• This does not satisfy the equation \(676 \neq 725\).

Therefore, the correct pair of length and breadth of the rectangle inscribed in the circle of radius 13 cm is 24 cm and 10 cm.