Finding the Distance from the Center of a Circle to a Chord

Imagine a circle with a diameter of 30 centimeters and a chord of 18 centimeters. How can we determine the distance from the center of the circle to this chord? In this article, we'll explore three different methods to solve this problem.

Method 1: Using a Diagram and Right Triangle

First, let's draw a diagram. If we draw a chord and a radius to one end of the chord, and then the apothem (perpendicular from the center to the chord) to bisect the chord, we create a right triangle. The hypotenuse of this right triangle is the radius of the circle, which is 15 cm (half of the diameter). One leg of the triangle is half the length of the chord, which is 9 cm (18 cm / 2).

Using the Pythagorean Theorem, we can find the distance from the center of the circle to the chord. The formula is:

Distance2 Radius2 - Half Chord Length2

Substituting the values:

Distance2 152 - 92

Distance2 225 - 81

Distance2 144

Distance √144 12 cm

Therefore, the distance from the center of the circle to the chord is 12 cm.

Method 2: Solving a Quadratic Equation

In this method, we use the Pythagorean Theorem with a quadratic equation. Let's denote the distance from the center of the circle to the midpoint of the chord as x. The length of the entire chord is 18 cm, so the leg of the right triangle formed by half the chord is 9 cm. The radius of the circle is 15 cm.

Using the Pythagorean Theorem:

(152 - x2) 92

225 - x2 81

x2 225 - 81

x2 144

x √144 12 cm

The same value is obtained, confirming that the distance from the center to the chord is 12 cm.

Method 3: Direct Application of the Chord Theorem

Using the chord theorem, which states that the product of the lengths of the segments of a secant or chord through two points of tangency to a circle is equal, we can also solve the problem. Let's denote the distance from the center of the circle to the chord as x. Since the chord bisects the distance from the center, we can set up the equation using the Pythagorean Theorem directly:

x2 92 152

x2 225 - 81

x2 144

x √144 12 cm

Thus, the distance from the center of the circle to the chord is 12 cm.

Additional Problem

Now, let's solve an additional problem: If the radius of a circle is 10 cm and the length of a chord is 16 cm, what is the distance of the chord from the center of the circle?

Following the same process, we can draw a right triangle where the radius is the hypotenuse (10 cm), half the chord (8 cm) is one leg, and the distance from the center of the circle to the chord is the other leg. Using the Pythagorean Theorem:

x2 82 102

x2 64 100

x2 36

x √36 6 cm

Therefore, the distance from the center of the circle to the chord is 6 cm.

Conclusion

The distance from the center of a circle to a chord can be calculated using various methods, including the Pythagorean Theorem, solving quadratic equations, and direct application of the chord theorem. Each method adheres to the geometric principles of circles and triangles. Understanding these methods is crucial for solving similar problems in circle geometry.