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 circumference equal to the sum of the circumferences of the two circles.

Radius (r₁) of 1^st circle = 19 cm
Radius (r₂) or 2^nd circle = 9 cm
Let the radius of 3^rd circle be r.
Circumference of 1^st circle = 2πr₁ = 2π (19) = 38π
Circumference of 2^nd circle = 2πr₂ = 2π (9) = 18π
Circumference of 3^rd circle = 2πr
Given that,
Circumference of 3^rd circle = Circumference of 1^st circle + Circumference of 2^nd circle
2πr = 38π + 18π = 56π

Therefore, the radius of the circle which has circumference equal to the sum of the circumference of the given two circles is 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.

Radius (r₁) of 1^st circle = 8 cm
Radius (r₂) of 2^nd circle = 6 cm
Let the radius of 3^rd circle be r.
Area of 1^st circle

Area of 2^nd circle

Given that,
Area of 3^rd circle = Area of 1^st circle + Area of 2^nd circle

However, the radius cannot be negative. Therefore, the radius of the circle having area equal to the sum of the areas of the two circles is 10 cm.

Question 3:

Given figure depicts an archery target marked with its five scoring areas 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. 

Radius (r₁) of gold region (i.e., 1^st circle)

Given that each circle is 10.5 cm wider than the previous circle.
Therefore, radius (r₂) of 2^nd circle = 10.5 + 10.5
21 cm
Radius (r₃) of 3^rd circle = 21 + 10.5
= 31.5 cm
Radius (r₄) of 4^th circle = 31.5 + 10.5
= 42 cm
Radius (r₅) of 5^th circle = 42 + 10.5
= 52.5 cm
Area of gold region = Area of 1^st circle

Area of red region = Area of 2^nd circle − Area of 1^st circle

Area of blue region = Area of 3^rd circle − Area of 2^nd circle

Area of black region = Area of 4^th circle − Area of 3^rd circle

Area of white region = Area of 5^th circle − Area of 4^th circle

Therefore, areas of gold, red, blue, black, and white regions are 346.5 cm², 1039.5 cm², 1732.5 cm², 2425.5 cm², and 3118.5 cm² respectively.

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 traveling at a speed of 66 km per hour? 

Diameter of the wheel of the car = 80 cm
Radius (r) of the wheel of the car = 40 cm
Circumference of wheel = 2πr
= 2π (40) = 80π cm
Speed of car = 66 km/hour

Distance travelled by the car in 10 minutes
= 110000 × 10 = 1100000 cm
Let the number of revolutions of the wheel of the car be n.
n × Distance travelled in 1 revolution (i.e., circumference)
= Distance travelled in 10 minutes

Therefore, each wheel of the car will make 4375 revolutions.

Question 5:

Tick the correct answer 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

Let the radius of the circle be r.
Circumference of circle = 2πr
Area of circle = πr²
Given that, the circumference of the circle and the area of the circle are equal.
This implies 2πr = πr²
2 = r
Therefore, the radius of the circle is 2 units.
Hence, the correct answer is A