Q2. If the coordinates of the center of a circle are (0, 0) and it passes through the point (3, 4), what is the radius of the circle?
Solution:
Radius = distance from center to point = √[(3 - 0)² + (4 - 0)²] = √[9 + 16] = √25 = 5 units.
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Q3. In a coordinate plane, what is the equation of a circle with center at (3, -2) and radius 4?
Solution:
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
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Q4. What is the measure of the angle subtended by an arc of a circle at the center if the arc length is 10 units and the radius is 5 units?
Solution:
Arc length = rθ, where θ is in radians. Thus, 10 = 5θ, so θ = 2 radians. Converting to degrees: θ = 2 * (180/π) ≈ 114.6 degrees.
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Q5. If the coordinates of the center of a circle are (2, 3) and the radius is 4, what is the equation of the circle?
Solution:
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
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Q6. What is the length of the arc of a circle with a radius of 6 units that subtends an angle of 60 degrees at the center?
Solution:
The length of an arc is given by L = (θ/360) * 2πr. Here, L = (60/360) * 2π(6) = (1/6) * 12π = 2π units.
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Q7. Two triangles are similar. If the lengths of the sides of the first triangle are 3, 4, and 5 units, what are the lengths of the corresponding sides of the second triangle if the shortest side is 6 units?
Solution:
The ratio of the sides is 6/3 = 2. Therefore, the corresponding sides are 6*2, 4*2, and 5*2, which are 6, 8, and 10 units.
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Q8. If triangle ABC is similar to triangle DEF, and the lengths of sides AB and DE are 6 cm and 9 cm respectively, what is the ratio of the areas of the triangles?
Solution:
The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (6/9)² = (2/3)² = 4/9.
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Q9. If triangle ABC is similar to triangle DEF, and the lengths of sides AB and DE are 6 cm and 9 cm respectively, what is the ratio of their areas?
Solution:
The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (6/9)² = (2/3)² = 4/9.
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Q10. What is the length of the diameter of a circle if its area is 50π square units?
Solution:
Area = πr², so 50π = πr². Thus, r² = 50, r = √50 = 5√2. Diameter = 2r = 10√2 units.
