Q1. In a right triangle, if one leg is 3 units and the hypotenuse is 5 units, what is the length of the other leg?
Solution:
Using the Pythagorean theorem, a² + b² = c², we have 3² + b² = 5², which gives b² = 25 - 9 = 16, so b = 4 units.
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Q2. 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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Q3. 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.
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Q4. In coordinate geometry, what is the slope of the line passing through the points (1, 2) and (3, 6)?
Q5. 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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Q6. What is the length of the diagonal of a rectangle with a width of 6 units and a height of 8 units?
Solution:
Using the Pythagorean theorem, the diagonal d = √(width² + height²) = √(6² + 8²) = √(36 + 64) = √100 = 10 units.
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Q7. 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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Q8. 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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Q9. 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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Q10. 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.
