Question

A chord of circle radius 15cm subtends an angle of 60 degree find the area of corresponding major and minor segment of the circle (use pie = 3.14) ​

Answer 1

Answer:

Step-by-step explanation:

Radius of the circle = 15 cm

ΔAOB is isosceles as two sides are equal.

∴ ∠A = ∠B

Sum of all angles of triangle = 180°

∠A + ∠B + ∠C = 180°

⇒ 2 ∠A = 180° - 60°

⇒ ∠A = 120°/2

⇒ ∠A = 60°

Triangle is equilateral as ∠A = ∠B = ∠C = 60°

∴ OA = OB = AB = 15 cm

Area of equilateral ΔAOB = √3/4 × (OA)2 = √3/4 × 152

= (225√3)/4 cm2 = 97.3 cm2

Angle subtend at the centre by minor segment = 60°

Area of Minor sector making angle 60° = (60°/360°) × π r2 cm2

= (1/6) × 152 π  cm2

=  225/6 π  cm2

=  (225/6) × 3.14 cm2 = 117.75  cm2

Area of the minor segment = Area of Minor sector - Area of equilateral ΔAOB

= 117.75  cm2 - 97.3 cm2 = 20.4 cm2

Angle made by Major sector = 360° - 60° = 300°

Area of the sector making angle 300° = (300°/360°) × π r2 cm2

= (5/6) × 152 π  cm2 =  1125/6 π  cm2

                                                 

Area of major segment = Area of Minor sector + Area of equilateral ΔAOB

= 588.75  cm2 + 97.3 cm2 = 686.05 cm2

#answerwithquality #BAL

Answer 2

${\bold{\underline{\boxed{Answer=686.06\:cm^{2}}}}}$

Solution : Radius (r) of the circle = 15 cm

Area of sector OPRQ = $\dfrac{60°}{360°}\times\pi\:r^{2}$

= $\dfrac{60°}{360°}\times3.14\times(15)^{2}$

= $\sf\cancel\dfrac{60°}{360°}\times3.14\times225=\dfrac{1}{6}\times706.5$

= 117.75 cm²

In ∆OPQ

⇒ ∠OPQ = ∠OQP (As OP = OQ)

⇒ ∠OPQ + ∠OQP + ∠POQ = 180°

⇒ 2 ∠OPQ = 120°

⇒ ∠OPQ = 120°/2

⇒ ∠OPQ = 60°

∆OPQ is an equilateral triangle.

⇒ ∴ Area of ∆OPQ = $\dfrac{\sqrt{3}}{4}\times(side)^{2}$

⇒$\dfrac{\sqrt{3}}{4}\times(15)^{2}=\dfrac{225\sqrt{3}}{4}\:cm^{2}$

⇒56.25√3

⇒97.312 cm²

⇒ Now, Area of Minor segment PRQ = Area of sector OPRQ – Area of ∆OPQ

⇒ 117.75 ― 97.3125

⇒ 20.437 cm²

$\implies$ Area of Major segment PSQ = Area of circle ― Area of segment PRQ

$\pi(15)^{2}-20.437$

⇒ 3.14 × 225 ― 20.437

⇒ 706.5 ― 20.437

⇒ 686.06 cm²