Circle Theorem: Angle at Center vs Circumference
The relationship between the angle subtended by a chord at the center of a circle and the angle subtended at the circumference is defined by a key circle theorem.
Angle at the Center Theorem
The angle subtended by an arc or a chord at the center of the circle is twice the angle subtended by the same arc or chord at any point on the circumference. Importantly, this applies when the points are on the same side of the chord relative to the center.
Mathematically, if $\theta_c$ is the angle at the center and $\theta_a$ is the angle at the circumference:
$ \theta_c = 2 \times \theta_a $Or conversely:
$ \theta_a = \frac{\theta_c}{2} $Applying the Theorem
Given:
- Angle subtended at the center ($\theta_c$) = $75^{\circ}$
We need to find the angle subtended at the circumference on the same side ($\theta_a$).
Using the theorem:
$ \theta_a = \frac{\theta_c}{2} $ $ \theta_a = \frac{75^{\circ}}{2} $ $ \theta_a = 37.5^{\circ} $Conclusion
Therefore, the angle subtended by the chord at the circumference on the same side is $37.5^{\circ}$.
Similar Questions
- A circle that has a radius of 5 centimeters, and there is a chord measuring 8 centimeters in length. What is the distance from the center of the circle to the chord?
- A circle of radius of 12 cm, two radii are drawn such that the length of the chord connecting their endpoints is 12 cm. What is the area of the minor segment formed?
- The line segment from the center of a circle to the midpoint of a chord is 10 cm long. If the radius is 26 cm, what is the length of the chord?
- A chord of a circle subtends an angle of 60° at the center. What is the angle subtended at a point on the circle in the same segment?
- In a circle, there is a chord measuring $20\text{ cm}$ that is located $15\text{ cm}$ away from the center. What is the approximate radius of the circle?
- A tangent is drawn to a circle with a radius of $5\text{ cm}$. At the point of tangency, what is the angle between the tangent and the radius?
- The radii of three concentric circles are in arithmetic progression. If the innermost radius is 7 cm and the outermost is 21 cm, what is the radius of the middle circle?
- A chord subtends an angle of 70° at the center of a circle. What is the measure of the angle subtended by the same chord at any point on the remaining part of the circle?
- Chord AB and chord CD are equal and subtend angles of $50^\circ$ and $x^\circ$ at the center respectively. Find $x$
- A chord is drawn in a circle with a radius of 15 cm. If the distance of the chord from the center is 9 cm, what is the length of the chord?
Important Questions from Circles, Chords and Tangents
If a tangent to a circle from a point P meets the circle at A with AP = 15 cm. Given that the radius of the circle is 8 cm, find the distance of P from the centre of the circle.
In a circle with a radius of 10 cm. XY and PQ are two parallel chords 12 cm and 16 cm in length, respectively. The two chords are situated on the opposite sides of the centre. The distance between the chords is:
Find the equation of the tangents to the circle x2 + y2 = 9 at x = 2.
If a chord of length 24 cm is at a distance of 5 cm from centre, then find the radius of the circle.
Let \( C_1 \) and \( C_2 \) be two circles which do not externally touch and intersect each other and \( O_1 \), and \( O_2 \) be the centers of the circles, respectively. Let AB be the common transverse tangent to the circles such that P, Q are the points of tangency respectively to \( C_1 \), \( C_2 \). Let R be the point of intersection of \( O_1 O_2 \) and AB. If \( \angle PO_1R = 60^\circ \), find \( \angle QO_2R \) and \( \angle QRO_2 \) respectively.
