Similarity and Trigonometry Basics

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Q1. What is the area of a sector of a circle with a radius of 5 cm and a central angle of 90 degrees?

Solution:

Area of sector = (θ/360) * πr² = (90/360) * π(5)² = (1/4) * 25π ≈ 6.25 cm².

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Q2. If two circles have radii of 4 cm and 6 cm, what is the ratio of their areas?

Solution:

Area ratio = (r1^2):(r2^2) = (4^2):(6^2) = 16:36 = 4:9.

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Q3. In a circle with a radius of 10 cm, what is the length of an arc that subtends a central angle of 60 degrees?

Solution:

Arc length = (θ/360) * 2πr = (60/360) * 2π(10) = (1/6) * 20π = 10.47 cm.

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Q4. If the diameter of a circle is 20 cm, what is the radius?

Solution:

Radius = diameter/2 = 20/2 = 10 cm.

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Q5. If the radius of a circle is doubled, by what factor does the area increase?

Solution:

Area = πr^2. If radius is doubled, new area = π(2r)^2 = 4πr^2, so area increases by a factor of 4.

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Q6. What is the length of a chord in a circle of radius 8 cm that subtends a central angle of 90 degrees?

Solution:

Chord length = 2r * sin(θ/2) = 2 * 8 * sin(45°) = 8√2 cm.

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Q7. What is the length of the diameter of a circle if its circumference is 31.4 cm?

Solution:

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

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Q8. A triangle is inscribed in a circle of radius 5 cm. What is the maximum area of the triangle?

Solution:

Maximum area = (1/2) * r^2 * sin(θ) = (1/2) * 5^2 * 1 = 25 cm².