A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment of the circle.
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Step-by-Step Solution
Step 1: Identify Given Values
We are given the radius of the circle, $r = 14 \text{ cm}$, and the angle subtended by the chord at the centre, $\theta = 60^\circ$. These are the key values needed for our calculations.
Step 2: Calculate Area of Sector
The area of a sector with angle $\theta$ and radius $r$ is given by the formula $\frac{\theta}{360^\circ} \times \pi r^2$. Substituting the given values, we get $\frac{60^\circ}{360^\circ} \times \frac{22}{7} \times (14)^2$.
Step 3: Simplify Area of Sector
Simplifying the expression, $\frac{60^\circ}{360^\circ}$ becomes $\frac{1}{6}$. Then, we multiply $\frac{1}{6}$ by $\frac{22}{7}$ and $196$. This simplifies to $\frac{308}{3} \text{ cm}^2$.
Step 4: Calculate Area of Triangle
The area of the triangle formed by the two radii and the chord can be calculated using the formula $\frac{1}{2} r^2 \sin\theta$. Substituting the values, we get $\frac{1}{2} (14)^2 \sin(60^\circ)$. We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$.
Step 5: Simplify Area of Triangle
Simplifying the expression for the area of the triangle, we multiply $\frac{1}{2}$ by $196$ and then by $\frac{\sqrt{3}}{2}$. This results in $49\sqrt{3} \text{ cm}^2$.
Step 6: Calculate Area of Minor Segment
The area of the minor segment is found by subtracting the area of the triangle from the area of the sector. So, we subtract $49\sqrt{3}$ from $\frac{308}{3}$.