Circle Angle Theorem Explained
The question relates the angle subtended by a chord at the center of a circle to the angle subtended by the same chord at a point on the circumference. We use a fundamental theorem of circles for this.
Angle Subtended by Chord Theorem
- Theorem Statement: The angle subtended by an arc or a chord at the center of a circle is double the angle subtended by it at any point on the remaining part of the circle.
- In mathematical terms: Angle at Center = $2 \times$ Angle at Circumference
Applying the Theorem
We are given the angle subtended by chord AB at the center.
- Angle at Center ($\angle AOB$) = $80^\circ$
- We need to find the angle $\angle ACB$, where C is a point on the circumference.
Using the theorem:
Angle at Center = $2 \times \angle ACB$
$80^\circ = 2 \times \angle ACB$
Calculating $\angle ACB$
To find $\angle ACB$, we rearrange the equation:
$\angle ACB = \frac{80^\circ}{2}$
$\angle ACB = 40^\circ$
Conclusion
Therefore, the measure of angle $\angle ACB$ is $40^\circ$. This corresponds to Option A.
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 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?
- If the angle subtended by a chord at the center is $150^\circ$, what is the angle subtended at a point on the circle in the same segment?
- 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.
