Understanding the Radius of a Smaller Circle in Relation to a Larger Circle

In geometry, one of the fundamental concepts is understanding the properties and relationships between circles, especially when they are in contact with each other. This article will explore a scenario where two circles touch internally, focusing on determining the radius of the smaller circle. Let's break down the problem step-by-step, ensuring clarity and precision throughout the explanation.

Problem Statement

The distance between the centers of two circles touching internally is 5 cm. The radius of the larger circle is 17 cm. What is the radius of the smaller circle?

Solution

To solve this problem, we start with the given information:

The distance between the centers of the two circles touching internally is 5 cm. The radius of the larger circle, ( R_1 ), is 17 cm.

When two circles touch internally, the distance between their centers is equal to the difference between their radii. This relationship can be represented by the equation:

[ D R_1 - R_2 ]

where D is the distance between the centers, ( R_1 ) is the radius of the larger circle, and ( R_2 ) is the radius of the smaller circle. Substituting the given values, we get:

[ 5 17 - R_2 ]

By solving for ( R_2 ), the radius of the smaller circle, we perform the following steps:

Subtract 17 from both sides of the equation: 5 17 - R2 R2 17 - 5 R2 12

Therefore, the radius of the smaller circle is 12 cm.

Explanation

Imagine starting from the center of the larger circle and drawing a radius of 5 cm. This radius extends to the point where the center of the smaller circle is located. The distance from the center of the smaller circle to the far end of the radius of the larger circle is the difference between the larger and smaller circle's radii, which is 12 cm. This 12 cm is the radius of the smaller circle.

Another way to visualize this is to connect the "touching" point of the two circles to the center of the larger circle with a line segment. This segment is divided into two parts: one part is the given distance between the centers (5 cm), and the other part is the radius of the smaller circle (12 cm).

It's important to note that this problem is straightforward and doesn't involve complex geometric concepts. The internal tangency condition simplifies the problem and allows us to use the basic relationship between the radii of the circles.

Common Questions

1. What if the phrase "touching internally" means one circle is inside another?
In this context, the condition "touching internally" means that one circle is fully contained within the other. The distance between their centers is the difference in their radii, and the scenario can be solved similarly.

2. Can the circles be concentric?
In this scenario, the circles are not concentric. Concentric circles share the same center, which is not the case in the problem described.

3. Is the radius of the smaller circle always 12 cm?
No, the radius of the smaller circle is 12 cm in this specific example, but in other similar problems, the radius will vary based on the given distances and radii of the circles involved.

Conclusion

Understanding the relationship between the radii of circles that touch internally is a key concept in geometry. By applying simple arithmetic and geometric principles, we can efficiently solve problems involving such circles. The solution to this problem demonstrates how the difference in radii determines the radius of the smaller circle when the distance between their centers is known.