A Chord of a Circle of Radius 12 cm Subtends an Angle of 120 Degrees at the Centre
This problem involves finding the length of a chord within a circle, given its radius and the angle it subtends at the centre. We can solve this using basic geometry and trigonometry.
Understanding the Problem:
- Circle: We have a circle with a radius of 12 cm.
- Chord: A chord is a line segment that connects two points on the circumference of the circle.
- Subtended Angle: The angle formed at the centre of the circle by the two radii drawn to the endpoints of the chord is 120 degrees.
Solution:
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Visualize: Draw a circle with the centre labelled as 'O'. Draw a radius OA. Mark a point B on the circumference, such that angle AOB is 120 degrees. Join points A and B to form the chord AB.
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Equilateral Triangle: Since angle AOB is 120 degrees, angle OAB and angle OBA are each 30 degrees (sum of angles in a triangle is 180 degrees). This makes triangle OAB an isosceles triangle with two equal sides OA and OB (the radii), and a base AB. We can further divide this isosceles triangle into two congruent 30-60-90 right-angled triangles by drawing a perpendicular from O to AB, intersecting AB at point M.
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Trigonometry: In triangle OMA, we have:
- Angle OAM = 30 degrees
- OA = 12 cm (radius)
- AM = AB/2 (M bisects AB)
Using trigonometry, we can find AM:
- cos 30° = AM/OA
- √3/2 = AM/12
- AM = 12√3/2 = 6√3 cm
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Chord Length: Since AM = AB/2, then:
- AB = 2 * AM = 2 * 6√3 = 12√3 cm
Therefore, the length of the chord is 12√3 cm.
