Question

If a chord of a circle of radius 28 cm makes an angle of 90° at the centre,then the area of the major segment is ​

Answer 1

Step-by-step explanation:

Solution:

We use the formula for the area of a sector of the circle to solve the problem.

In a circle with radius r and the angle at the centre with degree measure θ,

(i) Area of the sector = θ/360° × πr2

(ii) Area of the segment = Area of the sector - Area of the corresponding triangle

Area of the right triangle = 1/2 × base × height

Let's draw a figure to visualize the area to be calculated.

Here, Radius, r = 10 cm, θ = 90°

Visually it’s clear from the figure that,

AB is the chord that subtends a right angle at the centre.

(i) Area of minor segment APB = Area of sector OAPB - Area of right triangle AOB

(ii) Area of major segment AQB = πr² - Area of minor segment APB

Area of the right triangle ΔAOB = 1/2 × OA × OB

(i) Area of minor segment APB = Area of sector OAPB - Area of right ΔAOB

= θ/360° × πr2 - 1/2 × OA × OB

= 90°/360° × πr2 - 1/2 × r × r

= 1/4 πr2 -1/2r2

= r2 (1/4 π - 1/2)

= r2 (3.14 - 2)/4

= (r2 × 1.14)/4

= (10 × 10 × 1.14)/4 cm² (Since radius r is given as 10 cm)

= 28.5 cm²

(ii) Area of major sector AOBQ = πr2 - Area of minor sector OAPB

= πr2 - θ/360° × πr2

= πr2 (1 - 90°/360°)

= 3.14 × (10 cm)2 × 3/4

= 235.5 cm2

Answer 2

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Pls refer to the attachment.