Two circles touch each other internally. The sum of their area is 130π cm² and the distance between their centers is 8 cm. Find the radii of the two circles.

To find the radii of the two circles that touch each other internally, we can use the information given about their areas and the distance between their centers.

Given Information

• Sum of areas: 130π cm²
• Distance between centers: 8 cm

Formulas to Use

The area of a circle is calculated using the formula:

A = πr²

Let the radii of the two circles be r₁ and r₂.

Setting Up Equations

From the sum of the areas, we have:

πr₁² + πr₂² = 130π

This simplifies to:

r₁² + r₂² = 130

Since the circles touch internally, the distance between their centers is:

r₁ - r₂ = 8

Solving the Equations

We now have two equations:

• r₁² + r₂² = 130
• r₁ - r₂ = 8

From the second equation, we can express r₁ in terms of r₂:

r₁ = r₂ + 8

Substituting this into the first equation:

(r₂ + 8)² + r₂² = 130

Expanding the equation:

r₂² + 16r₂ + 64 + r₂² = 130

This simplifies to:

2r₂² + 16r₂ - 66 = 0

Dividing the entire equation by 2 gives:

r₂² + 8r₂ - 33 = 0

Finding the Roots

Using the quadratic formula, r = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = 8, and c = -33:

r₂ = (-8 ± √(8² - 4 × 1 × -33)) / (2 × 1)

r₂ = (-8 ± √(64 + 132)) / 2

r₂ = (-8 ± √196) / 2

r₂ = (-8 ± 14) / 2

This gives us two possible values for r₂:

• r₂ = 3 cm (taking the positive root)
• r₂ = -11 cm (not valid since radius cannot be negative)

Finding r₁

Now, substituting r₂ = 3 cm back into the equation for r₁:

r₁ = r₂ + 8 = 3 + 8 = 11 cm

Final Results

The radii of the two circles are:

• r₁ = 11 cm
• r₂ = 3 cm