We are given the radius of the circle, r = 14 cm, and the angle subtended by the chord at the center, θ = 90^°. This angle is for the sector.

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 can calculate the area of the sector.

(a) A chord is subtending an angle of $90^\circ$ at the centre of a circle of radius 14 cm. Find the area of the corresponding minor segment of the circle. OR (b) Find the area of the shaded region if length of radius of each circle is 7 cm. Each circle touches the other two externally.

Answer: The area of the corresponding minor segment is $56\text{ cm}^2$ .

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 center, $\theta = 90^\circ$ . This angle is for the sector.

Step 2: Calculate the area of the 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 can calculate the area of the sector.

Step 3: Calculate the area of the triangle

The area of the triangle formed by the two radii and the chord can be calculated using the formula $\frac{1}{2} r^2 \sin\theta$ . Since $\theta = 90^\circ$ , $\sin(90^\circ) = 1$ .

Step 4: Calculate the area of the minor segment

The area of the minor segment is the difference between the area of the sector and the area of the triangle. We subtract the calculated area of the triangle from the area of the sector to get the final answer.

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(a) A chord is subtending an angle of 90∘90^\circ90∘ at the centre of a circle of radius 14 cm. Find the area of the corresponding minor segment of the circle. OR (b) Find the area of the shaded region if length of radius of each circle is 7 cm. Each circle touches the other two externally.