Calculating the Distance from the Center of a Circle to a Chord Using the Pythagorean Theorem

Understanding how to calculate the distance from the center of a circle to a chord is a fundamental concept in geometry, applicable in various real-world scenarios, from engineering to architecture. This article will guide you through the process using the Pythagorean theorem, followed by illustrations to enhance comprehension.

Problem Setup

Consider a circle with a diameter of 10 centimeters (cm) and a chord that is 6 cm long. We need to determine the distance from the center of the circle to this chord.

Step-by-Step Solution

Calculate the radius of the circle:

Radius Diameter / 2 10 cm / 2 5 cm

Visualize the problem: The distance from the center of the circle to the chord, the radius, and half the chord length form a right triangle.

Apply the Pythagorean theorem:

( text{Radius}^2 text{Distance to chord}^2 left(frac{text{Chord length}}{2}right)^2 )

( 5^2 x^2 3^2 )

( 25 x^2 9 )

( x^2 16 )

( x 4 text{ cm} )

Illustration of the Problem

Imagine a circle with a center O and a diameter of 10 cm. A chord AB is drawn inside the circle, and the length of this chord is 6 cm. Draw radii from the center O to the endpoints A and B of the chord, and then draw a perpendicular from O to the chord, which intersects AB at a right angle at point P.

In this right triangle OAP, the distance OP (the perpendicular from the center to the chord) is the unknown we need to find. The hypotenuse OA is the radius of the circle, which is 5 cm, and AP is half the length of the chord, which is 3 cm.

Using the Pythagorean theorem:

( 5^2 OP^2 3^2 )

( 25 OP^2 9 )

( OP^2 16 )

( OP 4 text{ cm} )

Alternative Problem

In another scenario, given a circle with a radius of 9 cm and a chord 6 cm away from the center, we seek to find the length of the chord.

Step-by-Step Solution

Identify the known parameters:

Radius (R) 9 cm, Distance from the center to the chord (D) 6 cm.

Note that the distance from the center to the chord is the height of the two right triangles formed by the radius and the chord.

Apply the Pythagorean theorem to find half the chord length:

( text{Length of half the chord} sqrt{R^2 - D^2} )

( text{Length of half the chord} sqrt{9^2 - 6^2} sqrt{81 - 36} sqrt{45} 3sqrt{5} text{ cm} )

Calculate the full length of the chord:

( text{Chord Length} 2 times text{half the chord length} 2 times 3sqrt{5} 6sqrt{5} text{ cm} )

Additional Illustration

Imagine a circle with center O and a radius of 9 cm. A chord AB is 6 cm away from the center O. Draw the radius OA and OB, and from O, draw a perpendicular to AB, which intersects AB at a right angle at point P. Point P is 6 cm from O and divides AB into two equal halves.

Using the Pythagorean theorem:

( 9^2 6^2 left(frac{text{Chord length}}{2}right)^2 )

( 81 36 left(frac{text{Chord length}}{2}right)^2 )

( left(frac{text{Chord length}}{2}right)^2 45 )

( frac{text{Chord length}}{2} sqrt{45} 3sqrt{5} text{ cm} )

( text{Chord Length} 6sqrt{5} text{ cm} )

Approximating the chord length:

( 6sqrt{5} approx 6 times 2.236 13.416 text{ cm} )

Conclusion

Through the examples shown, we have demonstrated the application of the Pythagorean theorem to a variety of problems related to circles and chords. Understanding these principles can be instrumental in solving more complex geometric problems and can be particularly useful in fields requiring precise spatial awareness, such as engineering and architecture.