# A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector. (Use π = 3.14)

**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° × πr^{2}

(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° × πr^{2} - 1/2 × OA × OB

= 90°/360° × πr^{2} - 1/2 × r × r

= 1/4 πr^{2} -1/2r^{2}

= r^{2} (1/4 π - 1/2)

= r^{2 }(3.14 - 2)/4

= (r^{2} × 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 = πr^{2} - Area of minor sector OAPB

= πr^{2} - θ/360° × πr^{2}

= πr^{2} (1 - 90°/360°)

= 3.14 × (10 cm)^{2 }× 3/4

= 235.5 cm^{2}

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 12

**Video Solution:**

## A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector

NCERT Solutions Class 10 Maths Chapter 12 Exercise 12.2 Question 4

**Summary:**

The area of the minor segment APB and the area of major sector AOBQ if a chord of a circle of radius 10 cm subtends a right angle at the centre are 28.5 cm^{2} and 235.5 cm^{2} respectively.

**☛ Related Questions:**

- In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:(i) the length of the arc(ii) area of the sector formed by the arc(iii) area of the segment formed by the corresponding chord
- A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use π = 3.14 and √3 = 1.73)
- A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Use π = 3.14 and √3 = 1.73)
- A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see Fig. 12.11). Find(i) the area of that part of the field in which the horse can graze.(ii) the increase in the grazing area if the rope were 10 m long instead of 5 m.(Use π = 3.14)

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