Geometry: Angle at Center vs. Circumference
A fundamental theorem in circle geometry states that the angle subtended by an arc at the center of the circle is double the angle subtended by the same arc at any point on the circumference.
Applying the Angle Theorem
We are given the following information:
- The angle subtended by the chord at the center is $150^\circ$.
- We need to find the angle subtended by the same chord at a point on the circumference in the same segment.
Let the angle at the center be $\theta_c$ and the angle at the circumference be $\theta_{circ}$. According to the theorem:
$ \theta_c = 2 \times \theta_{circ} $
We can rearrange this formula to find the angle at the circumference:
$ \theta_{circ} = \frac{\theta_c}{2} $
Calculation
- Substitute the given angle at the center: $\theta_c = 150^\circ$.
- Calculate the angle at the circumference: $ \theta_{circ} = \frac{150^\circ}{2} $ $ \theta_{circ} = 75^\circ $
Conclusion
Therefore, the angle subtended by the chord at a point on the circle in the same segment is $75^\circ$.
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