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 $75^\circ$.
Similar Questions
- The number of triangles formed by diagonals from one vertex of an octagon is:
- The sum of all interior angles of a 20-sided polygon is:
- If the measure of an exterior angle of a regular polygon is $20^\circ$, determine the total number of its sides.
- What is the minimum distance between centers of two circles having radii 5 cm and 3 cm such that exactly three common tangents exist?
- In a circle, angle subtended by chord AB at the center is $80^\circ$. What is the measure of angle $\angle ACB$, where C lies on the circle?
- In a circle with center O, two chords, AB and CD, cross each other at right angles. if the distance from the center O to chord AB is 3 cm, while the distance from O to chord CD is 4 cm. What is the radius of the circle?
- A triangle ABC has a circle inscribed within it with sides a, b, c. The area of the triangle is A. What is the radius of the inscribed circle?
- In a triangle ABC, if the angle bisectors of $\angle B$ and $\angle C$ intersect at point I, and $\angle BIC = 110^\circ$, what is the measure of $\angle A$?
- In triangle PQR, ST is parallel to QR. If PS=5cm, SQ=10cm, and PT=6cm, find TR.
- In a triangle ABC, side BC is extended to a point D. The exterior angle at C is 105°. The interior angles at A and B are in the ratio 2:5. What is the measure of the largest interior angle of the triangle?
Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
