Question

in a circle of radius 10 cm an arc subtends a right angle at the centre. find (I) lenght of arc (ii) the area of segment formed by corresponding chord​

Answer 1

Answer:

ANSWER

In the mentioned figure,

O is the centre of circle,

AB is a chord

AXB is a major arc,

OA=OB= radius =21 cm

Arc AXB subtends an angle 60

o

at O.

i) Length of an arc AXB =

360

60

×2π×r

=

6

1

×2×

7

22

×21

=22cm

ii) Area of sector AOB =

360

60

×π×r

2

=

6

1

×

7

22

×(21)

2

=231cm

2

iii) Area of segment (Area of Shaded region) = Area of sector AOB− Area of △AOB

By trigonometry,

AC=21sin30

OC=21cos30

And, AB=2AC

∴ AB=42sin30=41×

2

1

=21 cm

∴ OC=21cos30=

2

21

3

cm

∴ Area of △ AOB =

2

1

×AB×OC

=

2

1

×21×

2

21

3

=

4

441

3

cm

2

∴ Area of segment (Area of Shaded region) =(231−

4

441

3

) cm

2

Answer 2

$\huge \fcolorbox{black}{red}{♛answer♛}$

Given :-

Radius of circle = 10 cm

Central angle, x = 90°

To find :-

(i) Length of arc

(ii)Area of corresponding Segment of chord of circle

Formula's used

$\small\bold\red{length \: of \: arc \: = 2\pi \: r \frac{x}{360} } \\ \small\bold\red{area \: of \: right \: angle \: triangle \: = \frac{1}{2} \times base \times height} \\ \small\bold\red{area \: of \: segment \: = area \: of \: sector \: - area \: of \: triangle} \\ \small\bold\red{area \: of \: sector \: = \pi \: {r}^{2} \frac{x}{360} }$

Solution :-

$\small\bold\red{(i) Length \: of \: arc = 2\pi \: r \frac{x}{360} } \\ \small\bold\red{ =2 \times \frac{22}{7} \times 10 \times \frac{90}{360} } \\ \small\bold\red{ = \frac{110}{7} = 15.714 \: cm}$

$\small\bold\red{Area \: of \: sector = \pi \: {r}^{2} \frac{x}{360} } \\ \small\bold\red{ = \frac{22}{7} \times 10 \times 10 \times \frac{90}{360} } \\ \small\bold\red{ = \frac{550}{7} {cm}^{2} } \\ \small\bold\red{Area \: of \:triangle \: = \frac{1}{2} \times 10 \times 10 = 50 {cm}^{2} } \\ \small\bold\red{Area \: of \:segment = area \: of \: sector \: - area \: of \: triangle} \\ \small\bold\red{ = \frac{550}{7} - 50 = \frac{550 - 350}{7} } \\ = \small\bold\red{ \frac{200}{7} = 28.571 {cm}^{2} }$