For part (a), we are given the radius of the circle, r = 7 cm, and the perimeter of the sector, which is 25 cm. We need to find the area of this sector.

The area of a sector can be calculated using the formula Area = (1)/(2) × r × l, where r is the radius and l is the arc length. Plugging in the values r=7 cm and l=11 cm, we get the area as 38.5 cm².

The area of a sector can be calculated using the formula Area = (1)/(2) × r × l, where r is the radius and l is the arc length. Plugging in the values r=7 cm and l=11 cm, we get the area as 38.5 cm².

For part (b), we are given the radius of the circle, r = 35 cm, and the angle subtended by the arc at the center, θ = 90^°. We need to find the area of the minor segment.

The area of a sector is given by the formula (θ)/(360^°) × π r². Substituting θ = 90^°, r = 35 cm, and π = (22)/(7), we calculate the area of the sector to be 962.5 cm².

Since the angle at the center is 90^°, the triangle formed by the two radii and the chord is a right-angled triangle. Its area can be calculated as (1)/(2) × base × height, which simplifies to (1)/(2) × r² × sinθ. With r=35 cm and θ=90^°, the area of the triangle is 612.5 cm².

Since the angle at the center is 90^°, the triangle formed by the two radii and the chord is a right-angled triangle. Its area can be calculated as (1)/(2) × base × height, which simplifies to (1)/(2) × r² × sinθ. With r=35 cm and θ=90^°, the area of the triangle is 612.5 cm².

(a) The perimeter of a sector of a circle of radius 7 cm is 25 cm. Find the area of the sector. OR (b) In a circle of radius 35 cm, an arc subtends an angle of $90^\circ$ at the centre. Find the area of the minor segment formed by the corresponding chord.

Answer: (a) The area of the sector is $38.5\text{ cm}^2$ . (b) The area of the minor segment is $350\text{ cm}^2$ .

Step-by-step solution

Step 1: Identify Given Information for Part (a)

For part (a), we are given the radius of the circle, $r = 7\text{ cm}$ , and the perimeter of the sector, which is $25\text{ cm}$ . We need to find the area of this sector.

Step 2: Calculate Arc Length for Part (a)

The perimeter of a sector is the sum of two radii and the arc length. Using the given values, we can find the arc length $l$ . Substituting $r=7$ cm and perimeter $=25$ cm, we get $25 = 2(7) + l$ , which simplifies to $l = 11$ cm.

Step 3: Calculate Area of Sector for Part (a)

The area of a sector can be calculated using the formula $\text{Area} = \frac{1}{2} \times r \times l$ , where $r$ is the radius and $l$ is the arc length. Plugging in the values $r=7$ cm and $l=11$ cm, we get the area as $38.5\text{ cm}^2$ .

Step 4: Identify Given Information for Part (b)

For part (b), we are given the radius of the circle, $r = 35\text{ cm}$ , and the angle subtended by the arc at the center, $\theta = 90^\circ$ . We need to find the area of the minor segment.

Step 5: Calculate Area of Sector for Part (b)

The area of a sector is given by the formula $\frac{\theta}{360^\circ} \times \pi r^2$ . Substituting $\theta = 90^\circ$ , $r = 35\text{ cm}$ , and $\pi = \frac{22}{7}$ , we calculate the area of the sector to be $962.5\text{ cm}^2$ .

Step 6: Calculate Area of Triangle for Part (b)

Since the angle at the center is $90^\circ$ , the triangle formed by the two radii and the chord is a right-angled triangle. Its area can be calculated as $\frac{1}{2} \times \text{base} \times \text{height}$ , which simplifies to $\frac{1}{2} \times r^2 \times \sin\theta$ . With $r=35$ cm and $\theta=90^\circ$ , the area of the triangle is $612.5\text{ cm}^2$ .

Step 7: Calculate Area of Minor Segment for Part (b)

The area of the minor segment is found by subtracting the area of the triangle from the area of the corresponding sector. Subtracting $612.5\text{ cm}^2$ (triangle area) from $962.5\text{ cm}^2$ (sector area) gives us $350\text{ cm}^2$ .

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(a) The perimeter of a sector of a circle of radius 7 cm is 25 cm. Find the area of the sector. OR (b) In a circle of radius 35 cm, an arc subtends an angle of 90∘90^\circ90∘ at the centre. Find the area of the minor segment formed by the corresponding chord.