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 cm, then the length of the arc \(XYZ\) is :
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Remember that \(45^{\circ}\) is exactly \(\frac{1}{8}\) of a full circle (\(360^{\circ}\)). So the arc length is simply \(\frac{1}{8}\) of the circumference (\(2\pi r\)).
To find the length of the arc \(XYZ\) of the circle subtending an angle of \(45^\circ\) at the center \(O\), we can use the formula for the arc length of a circle:
\(L = \frac{\theta}{360^\circ} \times 2 \pi r\)
where:
\(L\) is the arc length,
\(\theta\) is the angle subtended by the arc at the center,
\(r\) is the radius of the circle.
Given:
\(\theta = 45^\circ\)
\(r = 32 \, \text{cm}\)
Substitute these values into the formula:
\(L = \frac{45}{360} \times 2 \pi \times 32\)
Simplify:
\(L = \frac{1}{8} \times 2 \pi \times 32\)
\(L = \frac{64 \pi}{8}\)
\(L = 8 \pi \, \text{cm}\)
Therefore, the length of the arc \(XYZ\) is \(8 \pi \, \text{cm}\).
