Mathematics (School)

Algebra (Secondary) Arithmetic (Primary) Geometry Statistics & Probability Trigonometry

Q. What is the length of the altitude from the vertex of a triangle to the base if the base is 12 cm and the area is 36 cm²?

A. 3 cm
B. 6 cm
C. 9 cm
D. 12 cm

Solution

Area = 1/2 * base * height. Therefore, 36 = 1/2 * 12 * height. Solving gives height = 6 cm.

Correct Answer: B — 6 cm

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Q. What is the length of the altitude from the vertex of a triangle with a base of 10 cm and an area of 40 cm²?

A. 4 cm
B. 6 cm
C. 8 cm
D. 10 cm

Solution

Using the area formula: Area = 1/2 * base * height, we can solve for height: 40 = 1/2 * 10 * height, so height = 8 cm.

Correct Answer: B — 6 cm

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Q. What is the length of the altitude from the vertex of a triangle with a base of 12 cm and an area of 36 cm²?

A. 4 cm
B. 6 cm
C. 3 cm
D. 5 cm

Solution

The area of a triangle is given by Area = 1/2 * base * height. Rearranging gives height = 2 * Area / base = 2 * 36 / 12 = 6 cm.

Correct Answer: B — 6 cm

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Q. What is the length of the altitude from the vertex of a triangle with a base of 10 cm and an area of 30 cm²?

A. 3 cm
B. 5 cm
C. 6 cm
D. 8 cm

Solution

Using the area formula: Area = 1/2 * base * height, we can solve for height: 30 = 1/2 * 10 * height, so height = 30 / 5 = 6 cm.

Correct Answer: B — 5 cm

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Q. What is the length of the altitude from the vertex of a triangle with a base of 8 cm and an area of 32 cm²?

A. 4 cm
B. 6 cm
C. 8 cm
D. 10 cm

Solution

Using the area formula: Area = 1/2 * base * height, we can solve for height: 32 = 1/2 * 8 * height, thus height = 8 cm.

Correct Answer: A — 4 cm

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Q. What is the length of the altitude from the vertex opposite the base of 10 cm in an isosceles triangle with equal sides of 13 cm?

A. 12 cm
B. 10 cm
C. 9 cm
D. 8 cm

Solution

Using Pythagorean theorem: altitude = √(13² - (10/2)²) = √(169 - 25) = √144 = 12 cm.

Correct Answer: D — 8 cm

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Q. What is the length of the altitude from the vertex opposite the base of a triangle with a base of 10 cm and an area of 40 cm²?

A. 8 cm
B. 6 cm
C. 4 cm
D. 10 cm

Solution

Area = 1/2 * base * height. 40 = 1/2 * 10 * height. Height = (40 * 2) / 10 = 8 cm.

Correct Answer: B — 6 cm

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Q. What is the length of the altitude from vertex A to side BC in triangle ABC if AB = 10 cm, AC = 6 cm, and angle A = 30 degrees?

A. 3 cm
B. 5 cm
C. 6 cm
D. 10 cm

Solution

The altitude h = AC * sin(A) = 6 * sin(30°) = 6 * 0.5 = 3 cm.

Correct Answer: B — 5 cm

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Q. What is the length of the altitude from vertex A to side BC in triangle ABC if the area is 30 cm² and base BC = 10 cm?

A. 6 cm
B. 5 cm
C. 3 cm
D. 4 cm

Solution

Area = 1/2 * base * height. Therefore, 30 = 1/2 * 10 * height, which gives height = 6 cm.

Correct Answer: B — 5 cm

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Q. What is the length of the altitude from vertex A to side BC in triangle ABC with base BC = 12 cm and area = 36 cm²?

A. 6 cm
B. 4 cm
C. 3 cm
D. 5 cm

Solution

Area = 1/2 * base * height. Therefore, 36 = 1/2 * 12 * height, which gives height = 6 cm.

Correct Answer: A — 6 cm

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Q. What is the length of the altitude from vertex A(1, 1) to the base BC of triangle ABC with B(4, 1) and C(4, 5)?

A. 3
B. 4
C. 2
D. 5

Solution

The base BC is vertical, so the length of the altitude is the horizontal distance from A to line x = 4, which is |4 - 1| = 3.

Correct Answer: A — 3

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Q. What is the length of the altitude to the base of a triangle with an area of 50 cm² and a base of 10 cm?

A. 10 cm
B. 5 cm
C. 15 cm
D. 20 cm

Solution

Area = 1/2 * base * height. 50 = 1/2 * 10 * height. Height = (50 * 2) / 10 = 10 cm.

Correct Answer: B — 5 cm

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Q. What is the length of the arc of a circle with a radius of 10 cm that subtends an angle of 60 degrees at the center?

A. 10.47 cm
B. 6.28 cm
C. 17.45 cm
D. 10.00 cm

Solution

The length of an arc is given by the formula L = (θ/360) * 2πr. Here, L = (60/360) * 2π(10) = (1/6) * 20π ≈ 10.47 cm.

Correct Answer: A — 10.47 cm

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Q. What is the length of the arc of a circle with a radius of 10 units that subtends an angle of 60 degrees at the center?

A. 10π/3 units
B. 5π units
C. 10π/6 units
D. 20π/3 units

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2π(10) = (1/6) * 20π = 10π/3 units.

Correct Answer: A — 10π/3 units

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Q. What is the length of the arc of a circle with a radius of 4 cm and a central angle of 90 degrees?

A. 3.14 cm
B. 6.28 cm
C. 12.56 cm
D. 1.57 cm

Solution

Arc length = (θ/360) * 2πr = (90/360) * 2π(4) = (1/4) * 8π = 2π ≈ 6.28 cm.

Correct Answer: A — 3.14 cm

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Q. What is the length of the arc of a circle with a radius of 4 units and a central angle of 90 degrees?

A. 2π units
B. 4π units
C. π units
D. 8 units

Solution

Arc length = (θ/360) * 2πr = (90/360) * 2π(4) = (1/4) * 8π = 2π units.

Correct Answer: A — 2π units

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Q. What is the length of the arc of a circle with a radius of 5 cm and a central angle of 90°?

A. 3.93 cm
B. 7.85 cm
C. 12.57 cm
D. 15.71 cm

Solution

Arc length = (θ/360) * 2πr = (90/360) * 2π * 5 = (1/4) * 10π ≈ 7.85 cm.

Correct Answer: B — 7.85 cm

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Q. What is the length of the arc of a circle with a radius of 5 cm and a central angle of 60°?

A. 5.24 cm
B. 3.14 cm
C. 5.00 cm
D. 10.47 cm

Solution

Arc length = (θ/360) * 2 * π * r = (60/360) * 2 * π * 5 = 5.24 cm.

Correct Answer: A — 5.24 cm

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Q. 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?

A. 2π units
B. 3π units
C. 4π units
D. 5π units

Solution

The length of an arc is given by L = (θ/360) * 2πr. Here, L = (60/360) * 2π(6) = (1/6) * 12π = 2π units.

Correct Answer: B — 3π units

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Q. What is the length of the arc of a circle with a radius of 7 cm that subtends an angle of 60 degrees at the center?

A. 7π/3
B. 14π/3
C. 7π/6
D. 14π/6

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2π(7) = (1/6) * 14π = 7π/3 cm.

Correct Answer: A — 7π/3

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Q. What is the length of the diagonal of a rectangle with a length of 12 cm and a width of 9 cm?

A. 15 cm
B. 18 cm
C. 21 cm
D. 24 cm

Solution

Using the Pythagorean theorem: d² = length² + width² = 12² + 9² = 144 + 81 = 225, so d = 15 cm.

Correct Answer: A — 15 cm

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Q. What is the length of the diagonal of a rectangle with a length of 12 cm and a width of 5 cm?

A. 13 cm
B. 14 cm
C. 15 cm
D. 16 cm

Solution

The length of the diagonal can be calculated using the Pythagorean theorem: d = √(length² + width²). Here, d = √(12² + 5²) = √(144 + 25) = √169 = 13 cm.

Correct Answer: A — 13 cm

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Q. What is the length of the diagonal of a rectangle with a length of 6 cm and a width of 8 cm?

A. 10 cm
B. 12 cm
C. 14 cm
D. 16 cm

Solution

Diagonal = √(length² + width²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

Correct Answer: A — 10 cm

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Q. What is the length of the diagonal of a rectangle with a length of 6 m and a width of 8 m?

A. 10 m
B. 12 m
C. 14 m
D. 16 m

Solution

Diagonal = √(length² + width²) = √(6² + 8²) = √(36 + 64) = √100 = 10 m.

Correct Answer: A — 10 m

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Q. What is the length of the diagonal of a rectangle with a length of 8 cm and a width of 6 cm?

A. 10 cm
B. 12 cm
C. 14 cm
D. 16 cm

Solution

The length of the diagonal of a rectangle can be calculated using the Pythagorean theorem: diagonal = √(length² + width²) = √(8² + 6²) = √(64 + 36) = √100 = 10 cm.

Correct Answer: A — 10 cm

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Q. What is the length of the diagonal of a rectangle with a width of 3 cm and a height of 4 cm?

A. 5 cm
B. 7 cm
C. 6 cm
D. 8 cm

Solution

The length of the diagonal can be found using the Pythagorean theorem: d = √(3² + 4²) = √(9 + 16) = √25 = 5 cm.

Correct Answer: A — 5 cm

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Q. What is the length of the diagonal of a rectangle with a width of 6 cm and a height of 8 cm?

A. 10 cm
B. 12 cm
C. 14 cm
D. 16 cm

Solution

Diagonal = √(width² + height²) = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

Correct Answer: A — 10 cm

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Q. What is the length of the diagonal of a rectangle with a width of 6 units and a height of 8 units?

A. 10 units
B. 12 units
C. 14 units
D. 16 units

Solution

Using the Pythagorean theorem, the diagonal d = √(width² + height²) = √(6² + 8²) = √(36 + 64) = √100 = 10 units.

Correct Answer: A — 10 units

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Q. What is the length of the diagonal of a rectangle with length 12 cm and width 5 cm?

A. 13 cm
B. 14 cm
C. 15 cm
D. 16 cm

Solution

Using Pythagorean theorem: diagonal = √(length² + width²) = √(12² + 5²) = √(144 + 25) = √169 = 13 cm.

Correct Answer: A — 13 cm

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Q. What is the length of the diagonal of a rectangle with length 12 cm and width 9 cm?

A. 15 cm
B. 18 cm
C. 21 cm
D. 24 cm

Solution

Using the Pythagorean theorem: d² = length² + width² = 12² + 9² = 144 + 81 = 225, so d = 15 cm.

Correct Answer: A — 15 cm

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