(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∘ at the centre. Find the area of the minor segment formed by the corresponding chord.
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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 cm, and the perimeter of the sector, which is 25 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 Area=21×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.
Step 4: Identify Given Information for Part (b)
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.
Step 5: Calculate Area of Sector for Part (b)
The area of a sector is given by the formula 360∘θ×πr2. Substituting θ=90∘, r=35 cm, and π=722, we calculate the area of the sector to be 962.5 cm2.
Step 6: Calculate Area of Triangle for Part (b)
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 21×base×height, which simplifies to 21×r2×sinθ. With r=35 cm and θ=90∘, the area of the triangle is 612.5 cm2.
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 cm2 (triangle area) from 962.5 cm2 (sector area) gives us 350 cm2.