Finding the Measure of Angle AOB in a Circle with Given Radius and Chord

In geometric problems involving circles, the relationship between the radius, chord, and the angle subtended at the center can be explored using various trigonometric and geometric principles. This article delves into calculating the angle AOB in a circle given a specific radius and chord length, providing a detailed step-by-step solution. By the end of this article, readers will understand how to apply the Pythagorean theorem and the cosine rule to find the central angle of a given circle and chord.

Given Parameters

Circle's radius: 8 cm Chord length: 13 cm Circle's center: O Chord endpoints: A and B

Step-by-Step Solution

The first step is to find the distance from the center O to the chord AB. This distance is also known as the perpendicular distance from the center to the chord.

1. Finding the Perpendicular Distance (OM)

Step 1.1: Find the midpoint of chord AB. Let M be the midpoint. Thus, the length AM is half of the chord length. AM AB / 2 13 / 2 6.5 cm Step 1.2: Use the Pythagorean theorem in triangle OMA to find the distance OM. OM^2 AM^2 OA^2 OM^2 6.5^2 8^2 OM^2 42.25 64 OM^2 64 - 42.25 21.75 OM sqrt{21.75} ≈ 4.66 cm

2. Applying the Cosine Rule to Triangle OMA

Step 2.1: The cosine rule states that AB^2 OA^2 OB^2 - 2 cdot OA cdot OB cdot cos( angle AOB). 13^2 8^2 8^2 - 2 cdot 8 cdot 8 cdot cos( angle AOB) 169 64 64 - 128 cdot cos( angle AOB) 169 128 - 128 cdot cos( angle AOB) 128 cdot cos( angle AOB) 128 - 169 128 cdot cos( angle AOB) -41 cos( angle AOB) -41 / 128 angle AOB arccos(-41 / 128) angle AOB ≈ 2.36 radians or ≈ 135.3°

Alternative Approach

Another method to find the angle is to use the sine rule:

Step 3.1: In the right-angled triangle formed by the radius, the perpendicular distance, and half the chord, the sine of the angle at the center is the ratio of half the chord to the radius. sin( angle / 2) (13/2) / 8 13/16 0.8125 angle / 2 arcsin(0.8125) ≈ 54.34° angle 108.68°

Both methods yield similar results, confirming the angle AOB to be approximately 135.3 degrees or 2.36 radians.

Conclusion

This article has demonstrated how to find the measure of angle AOB in a circle given the radius and the length of a chord. Utilizing the Pythagorean theorem and the cosine rule provides a comprehensive approach to solving such geometric problems.

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