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Chapter 1:Real Numbers

Chapter 2:Polynomials

Chapter 3:Pair of Linear Equations in Two Variables

Chapter 4:Quadratic Equations

Chapter 5: Arithmetic Progression

Chapter 6: Triangles

Chapter 7:Coordinate Geometry

Chapter 8: Introduction to Trigonometry

Chapter 9:Some Applications of Trigonometry

Chapter 10: Circles

Chapter 11:Constructions

Chapter 12:Area Related to Circles

Chapter 13:Surface Areas and Volumes

Chapter 14:Statistics

Chapter 15:Probability

Chapter 12

Area Related to Circles

Exercise 12.1

Question 1
The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has a circumference equal to the sum of the circumferences of the two circles.

Answer

r1= 19 cm

Circumference of the 1cst circle = 2π×19 = 38π cm

r2 = 9 cm

Circumference of the 2nd circle = 2π×9 = 18π cm

Now, let the radius of new circle circle = R

According to question

circumference of new circle =Circumference of the 1st circle + Circumference of the 2nd circle

2πR= 38π+18π = 56π cm

2πR =56π

R= 56π/2π

R = 28 cm.

Question 2
The radii of two circles are 8 cm and 6 cm, respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

Answer

r1 = 8 cm

Area of 1st circle = π(8)2 = 64π

Radius of 2nd circle r2 = 6 cm

Area of 2nd circle = π(6)2 = 36π

So,

The sum of 1st and 2nd circle will be = 64π+36π = 100π

let the radius of new circle = R

Area of the new circle = πR2

The area of the new circle = Area of 1st circle + Area of 2nd circle

Or, πR2 = 100πcm2

R2 = 100cm2

So, R = 10cm

Question 3
. Fig. 12.3 depicts an archery target marked with its five scoring regions from the centre outwards as Gold, Red, Blue, Black and White. The diameter of the region representing Gold score is 21 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.

NCERT Ex-12.1 class 10
.
Answer

Diameter of circle =21cm

The radius of gold region r1 = 21/2 cm

Area of gold region = π r12 = π(10.5)2 = 346.5 cm2

So, the radius of red region r2 = 10.5cm+10.5cm = 21 cm

Thus,

∴ Area of red region = Area of 2nd circle − Area of gold region = (πr22−346.5) cm2

= (π(21)2 − 346.5) cm2

= 1386 − 346.5

= 1039.5 cm2

Similarly,

The radius of blue region r3 = 21 cm+10.5 cm = 31.5 cm

∴ Area of blue region = Area of third circle – Area of second circle

= π(31.5)2 – 1386 cm2

= 3118.5 – 1386 cm2

= 1732.5 cm2

The radius of black region r4 = 31.5 cm+10.5 cm = 42 cm

∴ Area of black region = Area of fourth circle – Area of third circle

= π(42)2 – 1386 cm2

= 5544 – 3118.5 cm2

= 2425.5 cm2

The Radius of white region r5 = 42 cm+10.5 cm = 52.5 cm

∴ Area of white region (n=5) = Area of fifth circle – Area of fourth circle

= π(52.5)2 – 5544 cm2

= 8662.5 – 5544 cm2

= 3118.5 cm2

Question 4
The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?

Answer

diameter of wheel d 1 = 80

The radius of car’s wheel = 80/2 = 40 cm

the circumference of wheels = 2πr = 80 π cm

Since speed of car =66kmh-1
66 X 1000 /60 = 1100 m min-1

Distance covered by the car in 1hr = (66×105) cm

In 10 minutes, the distance covered will be = (66×105×10)/60 = 1100000 cm/s

∴ Distance covered by car = 11×105 cm

Now, the no. of revolutions of the wheels = (Distance covered by the car/Circumference of the wheels)

=( 11×105)/80 π = 4375.

Question 5
tick the correct Solution: in the following and justify your choice : If the perimeter and the area of a circle are numerically equal, then the radius of the circle is

(A) 2 units (B)π units

(C) 4 units (D) 7 units

Answer

Since the perimeter of the circle = area of the circle,

2πr = πr2

Or, r = 2

So, option (A) is correct