Q1. A tangent to a circle is drawn from a point outside the circle. If the radius of the circle is 3 cm and the distance from the center to the point is 5 cm, what is the length of the tangent?
Solution:
Length of tangent = √(d² - r²) = √(5² - 3²) = √(25 - 9) = √16 = 4 cm.
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Q2. If a circle has an area of 36π cm², what is the radius of the circle?
Solution:
Area = πr², so 36π = πr², r² = 36, r = 6 cm.
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Q3. In a circle, if the angle subtended by an arc at the center is 80 degrees, what is the angle subtended at any point on the circumference?
Solution:
Angle at circumference = 1/2 * angle at center = 1/2 * 80 = 40 degrees.
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Q4. In a circle, if two chords AB and CD intersect at point E, and AE = 3 cm, EB = 4 cm, what is the length of segment CE if DE = 2 cm?
Solution:
Using the intersecting chords theorem: AE * EB = CE * DE, so 3 * 4 = CE * 2, CE = 6 cm.
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Q5. What is the area of a sector of a circle with a radius of 10 cm and a central angle of 90 degrees?
Solution:
Area of sector = (θ/360) * πr² = (90/360) * π(10)² = (1/4) * 100π = 25π cm².
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Q6. If a circle has an area of 36π cm², what is the radius?
Solution:
Area = πr², so 36π = πr², r² = 36, r = 6 cm.
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Q7. What is the area of a sector of a circle with a radius of 4 cm and a central angle of 60 degrees?
Solution:
Area of sector = (θ/360) * πr² = (60/360) * π(4)² = (1/6) * 16π = 8π/3 cm².
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Q8. Two circles intersect at points A and B. If the radius of the first circle is 4 cm and the second is 6 cm, what is the maximum distance between the centers of the circles?
Solution:
Maximum distance = r1 + r2 = 4 + 6 = 10 cm.
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Q9. What is the length of an arc of a circle with a radius of 3 cm that subtends an angle of 90 degrees at the center?
