Basic Geometric Concepts - Problems on Circles

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Q1. A circle is inscribed in a triangle with sides 8 cm, 15 cm, and 17 cm. What is the radius of the inscribed circle?

Solution:

Semi-perimeter = (8 + 15 + 17)/2 = 20 cm. Area = 60 cm². Radius = Area/semi-perimeter = 60/20 = 3 cm.

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Q2. What is the distance between the center of a circle at (2, 3) and a point on the circle at (5, 7)?

Solution:

Distance = √((5-2)² + (7-3)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

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Q3. If two circles have radii of 3 cm and 5 cm, what is the distance between their centers if they are externally tangent?

Solution:

Distance between centers = r1 + r2 = 3 + 5 = 8 cm.

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Q4. What is the length of the diameter of a circle with an area of 50π cm²?

Solution:

Area = πr²; 50π = πr²; r² = 50; r = √50; Diameter = 2r = 2√50 = 10 cm.

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Q5. If two circles have radii of 3 cm and 6 cm, what is the ratio of their areas?

Solution:

Area ratio = (r1²):(r2²) = (3²):(6²) = 9:36 = 1:4.

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Q6. A circle is inscribed in a square. If the side of the square is 8 cm, what is the radius of the circle?

Solution:

The radius of the inscribed circle is half the side of the square, so r = 8/2 = 4 cm.

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Q7. If a circle has an area of 36π cm², what is its radius?

Solution:

Area = πr²; 36π = πr²; r² = 36; r = 6 cm.

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Q8. A circle has a radius of 4 m. What is the length of an arc that subtends a central angle of 90 degrees?

Solution:

Arc length = (θ/360) * 2πr; θ = 90; Arc length = (90/360) * 2π(4) = π/2 m.

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Q9. What is the length of the diameter of a circle with a circumference of 31.4 cm?

Solution:

Circumference = πd, so d = C/π = 31.4/π = 10 cm.

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Q10. What is the area of a sector of a circle with a radius of 10 cm and a central angle of 60 degrees?

Solution:

Area of sector = (θ/360) * πr²; = (60/360) * π(10)² = (1/6) * 100π = 50π/3 cm².