The Radius of a Smaller Circle Internally Touched by a Larger Circle
Given a larger circle with a radius of 17 cm, and the distance between the centers of two circles when they touch internally is 5 cm, we need to find the radius of the smaller circle. This concept is often useful in various fields such as geometry, engineering, and architecture. Let's break down the solution step by step.
Understanding the Problem
The term 'touching internally' means that one circle lies wholly within the other, and they share a common point. The distance between the centers of the two circles is exactly 5 cm, which is the same as the difference between the radii of the two circles.
Using Geometric Properties
When two circles touch internally, the distance between their centers is equal to the difference in their radii. Therefore, if we let R be the radius of the larger circle and r be the radius of the smaller circle, we can write the equation:
[text{Distance between centers} R - r]In our case, the distance between the centers is 5 cm and the radius of the larger circle is 17 cm. So, we have:
[text{5 cm} 17 text{ cm} - r]Solving the Equation
Let's solve the equation to find the radius of the smaller circle:
begin{align*}5 17 - r r 17 - 5 r 12 text{ cm}end{align*}Thus, the radius of the smaller circle is 12 cm.
Visualization and Visualization
To better visualize this problem, imagine a larger circle with a radius of 17 cm, and a smaller circle with a radius of 12 cm, both sharing a common point (where they touch internally) and the line segment connecting their centers is 5 cm.
Conclusion
The key here is understanding the geometric relationship between the circles. Knowing that the distance between the centers is the difference between the radii allows us to set up and solve the equation easily.
This technique is not only useful in geometric problems but also in various real-world applications such as designing gears, creating sports equipment, or even in computer graphics.
So, the radius of the smaller circle is 12 cm.