If the sum of the areas of two circles with radii R₁ and R₂ is equal to the area of a circle of radius R, then
a. R₁ + R₂ = R
b. R₁² + R₂² = R²
c. R₁ + R₂ < R
d. R₁2 + R₂2 < R²
Solution:
Given, radii of two circles are R₁ and R₂
Sum of the areas of two circles is equal to the circumference of the circle with radius R.
We have to find the relation between the radii of the given circles.
Area of the circle = πr²
Now, area of circle with radius R₁ = πR₁²
Area of circle with radius R₂ = πR₂²
Sum of the areas = πR₁² + πR₂²
= π(R₁² + R₂²)
Area of circle with radius R = πR²
Given, π(R₁² + R₂²) = πR²
Cancelling out common term,
(R₁² + R₂²) = R²
Therefore, R₁² + R₂² = R²
✦ Try This: If the sum of the areas of two circles with radii 5 cm and 8 cm is equal to the area of a circle of radius R, then radius R is equal to
Given, area of two circles are 5 cm and 8 cm
We have to find the radius R.
Area of circle = πr²
Area of circle with radius 5 cm = π(5)²
= 25π
Area of circle with radius 8 cm = π(8)²
= 64π
Area of circle with radius R cm = πR²
Sum of areas of circle with radius 5 cm and 8 cm = 25π + 64π
= 89π
Given, 89π = πR²
R² = 89
Taking square root,
R = √89
R = 9.43 cm
Therefore, the radius R is 9.43 cm
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 12
NCERT Exemplar Class 10 Maths Exercise 11.1 Problem 1
If the sum of the areas of two circles with radii R₁ and R₂ is equal to the area of a circle of radius R, then a. R₁ + R₂ = R, b. R₁² + R₂² = R², c. R₁ + R₂ < R, d. R₁² + R₂² < R²
Summary:
If the sum of the areas of two circles with radii R₁ and R₂ is equal to the area of a circle of radius R, then R₁² + R₂² = R²
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