Question

In the given figure, O is the centre of circle. XYZ is an arc of the circle subtending an angle of $45^\circ$ at the centre. If the radius of the circle is $32 \text{ cm}$, then the length of the arc XYZ is :

• A. $4\pi \text{ cm}$
• B. $8\pi \text{ cm}$
• C. $64\pi \text{ cm}$
• D. $128\pi \text{ cm}$

Hint:

An angle of $45^\circ$ represents $\frac{45}{360} = \frac{1}{8}\text{th}$ of the entire circumference.
The total circumference of the circle is $2\pi r = 2\pi(32) = 64\pi \text{ cm}$.
The arc length is simply $\frac{64\pi}{8} = 8\pi \text{ cm}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks for the length of the arc XYZ of a circle.
We are given:
- The radius of the circle, $r = 32 \text{ cm}$
- The angle subtended by the arc at the centre, $\theta = 45^\circ$

Step 2: Key Formula or Approach:
The length of an arc ($l$) of a circle of radius $r$ which subtends an angle $\theta$ at the centre is given by:
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \]

Step 3: Detailed Explanation:

• Write down the known parameters:
- Angle, $\theta = 45^\circ$
- Radius, $r = 32 \text{ cm}$

• Substitute these values into the arc length formula:
\[ \text{Length of arc XYZ} = \frac{45^\circ}{360^\circ} \times 2 \times \pi \times 32 \]

• Simplify the fraction $\frac{45^\circ}{360^\circ}$:
Divide both numerator and denominator by 45:
\[ \frac{45}{360} = \frac{1}{8} \]

• Compute the remaining part of the expression:
\[ \text{Length of arc XYZ} = \frac{1}{8} \times 64\pi \]

• Perform the division:
\[ \text{Length of arc XYZ} = 8\pi \text{ cm} \]


Step 4: Final Answer:
The length of the arc XYZ is $8\pi \text{ cm}$.
Hence, option (B) is correct.