Q1. What is the measure of the central angle that subtends an arc of 120 degrees in a circle?
Solution:
The measure of the central angle is equal to the measure of the arc it subtends, which is 120 degrees.
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Q2. What is the relationship between the lengths of two tangents drawn from an external point to a circle?
Solution:
The lengths of two tangents drawn from an external point to a circle are equal.
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Q3. In a circle, if two angles subtended by the same arc are equal, what can be concluded about those angles?
Solution:
Angles subtended by the same arc at the circumference of a circle are equal.
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Q4. In a circle, if a radius is drawn to a point where a tangent touches the circle, what is the angle between the radius and the tangent?
Solution:
The radius drawn to the point of tangency is perpendicular to the tangent, forming a 90-degree angle.
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Q5. If a circle has a radius of 4 cm, what is the diameter of the circle?
Solution:
The diameter of a circle is twice the radius. For a radius of 4 cm, the diameter is 2 * 4 = 8 cm.
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Q6. If a circle has a radius of 4 cm, what is the area of the circle?
Solution:
The area A of a circle is given by A = πr². Therefore, A = π(4)² = 16π cm².
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Q7. In a circle, if the radius is 10 cm, what is the length of an arc that subtends a central angle of 60 degrees?
Solution:
The length of an arc is given by L = (θ/360) * 2πr. Here, L = (60/360) * 2π(10) = (1/6) * 20π ≈ 10.47 cm.
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Q8. If a tangent and a chord intersect at a point on the circle, what is the relationship between the angle formed and the angle subtended by the chord at the opposite arc?
Solution:
The angle formed by the tangent and the chord is equal to the angle subtended by the chord at the opposite arc.
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Q9. What is the relationship between the angles subtended by the same arc at the center and at any point on the circumference?
Solution:
The angle subtended at the center is always double the angle subtended at any point on the circumference.
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Q10. If a circle has a radius of 7 cm, what is the length of the circumference?
Solution:
The circumference of a circle is given by the formula C = 2πr. Therefore, C = 2π(7) = 14π cm.
