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Circle Area
Find the area of a circle whose radius is 7 m.
The area of a circle is calculated using the formula A = πr². With a radius of 7 m, the area is π × 7² = π × 49 ≈ 154 sq m, which corresponds to option B.
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Area Of Quadrilaterals
Find the area of a quadrilateral if one of its diagonals is 20 cm and its offsets are 9 cm and 6 cm.
The area of a quadrilateral given one diagonal and its offsets (perpendicular distances from the diagonal to the opposite vertices) is calculated as half the product of the diagonal and the sum of the offsets: Area = 1/2 × diagonal × (offset1 + offset2) = 1/2 × 20 × (9 + 6) = 150 cm².
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Geometry
A rectangular carpet has an area of 120 sq.m and a perimeter of 46 m. What is the length of its diagonal?
Let the length and width of the carpet be L and W. Given area = L × W = 120 and perimeter = 2(L + W) = 46, so L + W = 23. Solving these, we get L and W as roots of x^2 - 23x + 120 = 0, which are 15 and 8. The diagonal length = √(L² + W²) = √(15² + 8²) = √(225 + 64) = √289 = 17 m.
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Geometry
The perimeter of a semicircle is 144 cm. What is the radius?
The perimeter of a semicircle is given by the formula P = πr + 2r. Given P = 144 cm, solving 144 = r(π + 2) gives r ≈ 30 cm.
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Circle Area
If the radius of a circle is reduced by 50%, its area is reduced by how much?
The area of a circle is proportional to the square of its radius. If the radius is reduced by 50%, the new radius is 0.5 times the original. Therefore, the new area is (0.5)^2 = 0.25 times the original area, meaning the area is reduced by 75%. Hence, the correct answer is 75%.
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Circle Geometry
If the circumference of a circle is 352 meters, then its area in square meters is: ________?
Given the circumference C = 352 meters, we use the formula C = 2πr to find the radius r = C/(2π) = 352/(2×3.14) ≈ 56 meters. Then, the area A = πr² ≈ 3.14 × 56² = 3.14 × 3136 ≈ 9856 square meters, which matches option A.
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Triangles
The altitude of an equilateral triangle with side length 2√3 cm is ________?
The altitude of an equilateral triangle with side length s is given by (s√3)/2. Here, s = 2√3 cm, so altitude = (2√3 × √3)/2 = (2 × 3)/2 = 3 cm.
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Ratios and Proportions
If the ratio of the areas of two squares is 9:1, what is the ratio of their perimeters?
The area of a square is proportional to the square of its side length. Given the area ratio is 9:1, the side lengths ratio is the square root of 9:1, which is 3:1. Since the perimeter is directly proportional to the side length, the perimeter ratio is also 3:1.
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Triangles
What is the area of a triangle with side lengths a = 1 m, b = 2 m, and c = 3 m?
The given sides 1m, 2m, and 3m do not satisfy the triangle inequality (1 + 2 = 3), so these lengths cannot form a triangle. Therefore, the area is 0 square meters.
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Properties Of Rhombus
In a rhombus whose area is 144 sq.cm, one of its diagonals is twice as long as the other. The lengths of its diagonals are: _________?
The area of a rhombus is given by (1/2) × (diagonal1) × (diagonal2). Let the diagonals be d and 2d. Then the area = (1/2) × d × 2d = d². Given the area is 144 sq.cm, so d² = 144, which means d = 12 cm. Therefore, the diagonals are 12 cm and 24 cm. This matches option B exactly. Option C (6√2 cm, 12√2 cm) corresponds approximately to (8.49 cm, 16.97 cm), which does not satisfy the area condition. Hence, the correct answer is B.