Q. 23

# Find the differen

Given: Length of chord = 5 cm

Angle of sector = 90°

To find: Difference of areas of two segments.

Explanation:

Length of the chord = 5 cm (Given)

Let r be the radius of the circle.

Then, OA = OB = r cm

Now, angle subtended at the center of the sector OABO = 90°

Angle subtended at the center of the sector OABO (in radians) = θ = π/2

Triangle AOB is a right angled triangle.

So, by Pythagoras theorem, (AB)2 = (OA)2 + (OB)2

25 = 2r2

Also, AOB is an isosceles triangle.

Since, line segment OD is perpendicular on AB, therefore it divides Ab into two equal parts. Thus, AD = DB = 5/2 = 2.5 cm

So, in right angled triangle AOD, by Pythagoras theorem,

= 25/4

h = 5/2 = 2.5 cm

Area of the isosceles triangle AOB (1/2) × Base × Height

= (1/2) × 5 × (5/2) = 25/4 cm2

Now, Area of the minor sector = (1/2)r2θ

= = (25π/8) cm2

Area of the minor segment = Area of the minor sector – Area of the isosceles triangle

=

Area of the major segment = Area of the circle – Area of the minor segment

=

=

Difference of the areas of two segments of a circle =

|Area of major segment – Area of minor segment|

Hence, the required difference of the areas of two segments is

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