Distance from the Center of a Circle to a Chord: Solving with Geometry and the Pythagorean Theorem

In this article, we explore the problem of finding the distance from the center of a circle to a given chord. This is a classic problem in geometry, and it involves applying the Pythagorean theorem in a strategic manner. We will walk through multiple solutions, explore the underlying geometry, and present the problem in a way that is both educational and practical.

Introduction to the Problem

The problem at hand involves a chord of length 6 cm within a circle of radius 5 cm. Our goal is to determine the distance from the chord to the center of the circle. To solve this, we will visualize and utilize a series of geometric principles and the Pythagorean theorem.

Solving the Problem: A Step-by-Step Guide

The problem can be approached by forming a right-angled triangle. Let's denote the center of the circle as O and the endpoints of the chord as A and B.

Step 1: Join the center of the circle O with one of the endpoints of the chord, say A. This segment is equal to the radius of the circle, which is 5 cm.

Step 2: Join O with the other endpoint of the chord, B. Again, this is equal to the radius, so it is also 5 cm.

Step 3: Draw a line from the midpoint of the chord, M, to the center O. This line is perpendicular to the chord and will help us use the Pythagorean theorem.

Step 4: The length of the chord is given as 6 cm, so the distance from the center of the chord to the endpoints A and B is 3 cm (half of the chord).

Now we can draw a right-angled triangle with one leg being the distance from the center of the circle to the midpoint of the chord (let's call this x), the other leg being the segment from the midpoint of the chord to one of the endpoints of the chord (3 cm), and the hypotenuse being the radius of the circle (5 cm).

Step 5: Apply the Pythagorean theorem to this triangle:

x^2 3^2 5^2

x^2 9 25

x^2 16

x sqrt{16} 4 cm

Alternative Solutions

Several solvers have provided different but equally valid methods to solve this problem. One such solution is detailed below:

Solution by Red John

Red John used a clever method to solve the problem:

Step 1: Consider the right-angled triangle formed by the radius of the circle, half the chord, and the distance from the center of the circle to the chord.

Step 2: Use the Pythagorean theorem directly:

x^2 3.5^2 3.7^2

x^2 12.25 13.69

x^2 1.44

x sqrt{1.44} 1.2 cm

Using Circle Properties

Another solver used circle properties to solve the problem without explicitly applying the Pythagorean theorem. They noted:

Step 1: Use two circle properties: the fact that the diameter is a perpendicular bisector of the chord and the chord intersection theorem.

Step 2: Set up the equation using the chord intersection theorem:

3.7^2 - x^2 3.5^2

13.69 - x^2 12.25

x^2 1.44

x sqrt{1.44} 1.2 cm

Conclusion

In conclusion, the distance from the center of the circle to the chord is (1.2 text{ cm}). This problem demonstrates the power and versatility of the Pythagorean theorem and circle properties in solving geometric problems.

By understanding and applying these basic principles, you can solve a wide range of similar problems. The key is to visualize the problem correctly and apply the appropriate mathematical tools in a logical and systematic manner.