Q3. 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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Q4. 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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Q5. 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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Q6. 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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Q7. 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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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 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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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 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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Q10. 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.
