How to Calculate the Distance Between Two Circles with Their Common Tangent

To find the distance between the centers of two circles with given radii and the length of their common tangent, we can use specific formulas. Let's explore these calculations in detail.

When the Common Tangent is Given

Suppose we have two circles with radii 12 cm and 4 cm, and the length of their common tangent is 15 cm. We will use the following formula to find the distance d between the centers of these two circles:

d sqrt{L^2 r_1 cdot r_2}

Steps to Solve

Determine the given values: (L 15 , text{cm}) (r_1 12 , text{cm}) (r_2 4 , text{cm}) Calculate (r_1 cdot r_2): (r_1 cdot r_2 12 cdot 4 48 , text{cm}^2) Substitute the values into the formula: (d sqrt{15^2 48})

Calculation

Calculate (15^2): (15^2 225 , text{cm}^2) Construct the square root expression: (d sqrt{225 24} sqrt{249}) Finally, calculate the square root: (d approx 21.93 , text{cm})

Therefore, the distance between the centers of the two circles is approximately 21.93 cm.

Direct and Transverse Common Tangent

The previous formula assumes the tangent is direct. However, the problem does not specify whether the tangent is direct or transverse. Let's explore both cases.

Direct Tangent

For a direct common tangent, the distance (O_1O_2) between the centers of two circles with radii (R_1) and (R_2) and a common tangent (T) is given by:

(O_1O_2^2 T^2 (R_2 - R_1)^2)

Steps to Solve

Determine the given values: (T 15 , text{cm}) (R_1 12 , text{cm}) (R_2 4 , text{cm}) Subtract the radii: (R_2 - R_1 4 - 12 -8 , text{cm})

Calculation

Calculate ((R_2 - R_1)^2): ((8)^2 64 , text{cm}^2) Construct the formula: (O_1O_2^2 15^2 64 225 64 289 , text{cm}^2) Take the square root: (O_1O_2 sqrt{289} 17 , text{cm})

Thus, the distance between the centers of the two circles is 17 cm in the case of a direct tangent.

Transverse Tangent

A transverse tangent is more complex to handle and involves finding the external common tangents. However, based on the problem statement, we can assume the direct tangent case is the primary scenario.

Note: Solve for a transverse tangent would involve additional geometric properties and formulas, which are beyond the scope of this basic problem.

In conclusion, the distance between the centers of two circles given their radii and a common tangent can be calculated using specific formulas. The direct tangent case provides a straightforward calculation.