Finding Central Angle from Arc Length and Circumference
The problem asks for the central angle subtended by an arc (AB) given its length and the circle's circumference.
Key Information Provided:
- Arc Length (L) = 14 cm
- Circle Circumference (C) = 180 cm
Formula Used:
The ratio of the arc length to the total circumference is equal to the ratio of the central angle (\(\theta\)) to the total angle in a circle (360°).
\( \frac{\text{Arc Length}}{\text{Circumference}} = \frac{\text{Central Angle}}{360^\circ} \)
Using the given variables:
\( \frac{L}{C} = \frac{\theta}{360^\circ} \)
Calculation Steps:
- Substitute the known values into the formula: \( \frac{14 \text{ cm}}{180 \text{ cm}} = \frac{\theta}{360^\circ} \)
- Rearrange the equation to solve for the central angle \(\theta\): \( \theta = \frac{14}{180} \times 360^\circ \)
- Simplify the expression: \( \theta = 14 \times \left( \frac{360}{180} \right)^\circ \) \( \theta = 14 \times 2^\circ \) \( \theta = 28^\circ \)
Result:
The angle made by arc AB at the centre of the circle is 28°.
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