We are given the radius of the circle, r = 14 cm, and the angle subtended by the chord at the centre, θ = 60^°. These are the key values needed for our calculations.

The area of a sector with angle θ and radius r is given by the formula (θ)/(360^°) × π r². Substituting the given values, we get (60^°)/(360^°) × (22)/(7) × (14)².

Simplifying the expression, (60^°)/(360^°) becomes (1)/(6). Then, we multiply (1)/(6) by (22)/(7) and 196. This simplifies to (308)/(3) cm².

The area of the triangle formed by the two radii and the chord can be calculated using the formula (1)/(2) r² sinθ. Substituting the values, we get (1)/(2) (14)² sin(60^°). We know that sin(60^°) = (√3)/(2).

The area of the triangle formed by the two radii and the chord can be calculated using the formula (1)/(2) r² sinθ. Substituting the values, we get (1)/(2) (14)² sin(60^°). We know that sin(60^°) = (√3)/(2).

Simplifying the expression for the area of the triangle, we multiply (1)/(2) by 196 and then by (√3)/(2). This results in 49√3 cm².

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.

Answer: The area of the corresponding minor segment of the circle is $\left(\frac{308}{3} - 49\sqrt{3}\right) \text{ cm}^2$ .

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 $\frac{308}{3}$ .

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