Circles Problem Overview

This problem involves three circles that are touching each other externally. When circles touch externally, a key geometric principle is that the distance between their centers is equal to the sum of their radii. We are given the distances between the centers of these three circles, and our goal is to determine the individual radii of each circle.

Radii Definition and Setup

Let's define the radii of the three circles:

• Let \(r_1\) be the radius of the first circle.
• Let \(r_2\) be the radius of the second circle.
• Let \(r_3\) be the radius of the third circle.

Based on the information provided, the distances between the centers are 5 cm, 6 cm, and 7 cm. We can set up a system of linear equations using the principle that the distance between the centers of two externally touching circles is the sum of their radii:

1. Distance between center 1 and center 2: \(r_1 + r_2 = 5\) cm (Equation 1)
2. Distance between center 2 and center 3: \(r_2 + r_3 = 6\) cm (Equation 2)
3. Distance between center 1 and center 3: \(r_1 + r_3 = 7\) cm (Equation 3)

Solving Equations for Radii

To find the values of \(r_1\), \(r_2\), and \(r_3\), we can solve this system of three linear equations. A common method is to add all three equations together:

Divide the entire equation by 2:

Let's call this Equation 4.

Now, we can find each individual radius by subtracting the original equations (1, 2, 3) from Equation 4:

• To find \(r_3\): Subtract Equation 1 from Equation 4. \[ (r_1 + r_2 + r_3) - (r_1 + r_2) = 9 - 5 \] \[ r_3 = 4 \text{ cm} \]
• To find \(r_1\): Subtract Equation 2 from Equation 4. \[ (r_1 + r_2 + r_3) - (r_2 + r_3) = 9 - 6 \] \[ r_1 = 3 \text{ cm} \]
• To find \(r_2\): Subtract Equation 3 from Equation 4. \[ (r_1 + r_2 + r_3) - (r_1 + r_3) = 9 - 7 \] \[ r_2 = 2 \text{ cm} \]

Calculated Radii and Options Comparison

The radii of the three circles are 3 cm, 2 cm, and 4 cm. Arranging them in ascending order, the radii are 2 cm, 3 cm, and 4 cm.

Let's compare these calculated radii with the given options:

The calculated radii match Option 1 perfectly. This detailed step-by-step method ensures accuracy in finding the radii of circles touching externally given their center distances.