Determining the Distance of a Chord from the Center of a Circle
Problem Statement:
The radius of a circle is 25 cm and the length of one of its chords is 48 cm. What will be the distance of the chord from the center of the circle?
Step-by-Step Solution
Given:
Radius of the circle, ( R 25 ) cm Length of the chord, ( L 48 ) cmWe need to find the distance of the chord from the center of the circle, let's denote it as ( d ).
Consider the circle with center ( O ) and radius 25 cm. Draw a chord ( AB ) of length 48 cm. Join ( O ) to the midpoint of ( AB ), say ( M ). The line ( OM ) is perpendicular to ( AB ) and bisects it into two equal parts of 24 cm each.
We now have a right triangle ( OMA ) where:
OA is the radius of the circle, ( OA 25 ) cm AM is half of the chord, ( AM 24 ) cm OM is the distance from the center to the chord, ( OM d )Using the Pythagorean theorem in triangle ( OMA ):
[ OA^2 OM^2 AM^2 ]Substituting the known values:
[ 25^2 d^2 24^2 ] [ 625 d^2 576 ] [ d^2 625 - 576 ] [ d^2 49 ] [ d sqrt{49} 7 text{ cm} ]Hence, the distance of the chord from the center of the circle is 7 cm.
Explanation with Diagram
Figure 1: Circle with Radius 25 cm and Chord Length 48 cm
Draw a diameter of the circle of radius 25 cm. Draw a chord of length 48 cm parallel to the diameter. From the center of the circle, draw a line to the center of the chord, bisecting it into two equal parts of 24 cm each. Connect the endpoints of the chord to the center, forming two right triangles. Apply the Pythagorean theorem in one of these right triangles to find the distance from the center to the chord.Conclusion
The distance of the chord from the center of the circle is 7 cm. This distance is calculated based on the properties of right triangles and the Pythagorean theorem. By visualizing and applying these principles, the problem is easily solved.