A Chord Of A Circle Of Radius 14 Cm Subtends An Angle Of 120 Degree At The Centre

Suzhoumeite

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Aug 30, 2024

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Finding the Length of a Chord in a Circle

This article will guide you through calculating the length of a chord in a circle with a radius of 14 cm that subtends an angle of 120 degrees at the center.

Understanding the Problem

Imagine a circle with a radius of 14 cm. Now, draw a line segment connecting two points on the circle's circumference. This line segment is called a chord. The angle formed at the center of the circle by the two radii drawn to the endpoints of the chord is the central angle. In our case, this central angle is 120 degrees.

Using Geometry and Trigonometry

To find the length of the chord, we can use the following steps:

1. Divide the Central Angle: The central angle of 120 degrees divides the circle into three equal parts. Each part is an equilateral triangle with side lengths equal to the radius of the circle.

2. Isosceles Triangle Formation: The chord and two radii connecting its endpoints form an isosceles triangle. The central angle is the angle between the two radii.

3. Applying Law of Cosines: The Law of Cosines can be used to find the length of the chord (which is the base of the isosceles triangle). The formula is:

   c² = a² + b² - 2ab cos(C)
   

   where:

   • c is the length of the chord (the side opposite angle C)
   • a and b are the lengths of the radii (equal to 14 cm)
   • C is the central angle (120 degrees)

4. Calculation: Substitute the values into the formula:

   c² = 14² + 14² - 2(14)(14) cos(120°)
   c² = 196 + 196 - 392 * (-0.5) 
   c² = 588
   c = √588 = 14√3 cm
   

Conclusion

Therefore, the length of the chord in a circle of radius 14 cm that subtends an angle of 120 degrees at the center is 14√3 cm.