Basic Geometric Concepts - Coordinate Geometry for Exams

Basic Geometric Concepts - Coordinate Geometry Applications

Q. If a circle has a radius of 5, what is its area?

A. 25π
B. 10π
C. 20π
D. 15π

Solution

Area of a circle: A = πr² = π(5)² = 25π.

Correct Answer: A — 25π

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Q. If a line has a slope of -2 and passes through the point (3, 5), what is its y-intercept?

A. 1
B. 3
C. 5
D. 7

Solution

Using the point-slope form: y - y1 = m(x - x1) => y - 5 = -2(x - 3). Setting x = 0 gives y = 1.

Correct Answer: B — 3

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Q. If a line has a slope of -3 and passes through the point (2, 5), what is its equation in slope-intercept form?

A. y = -3x + 11
B. y = -3x + 5
C. y = 3x + 5
D. y = -3x + 2

Solution

Using y = mx + b: 5 = -3(2) + b => 5 = -6 + b => b = 11. Thus, y = -3x + 11.

Correct Answer: A — y = -3x + 11

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Q. What is the area of a rectangle with vertices at (1, 1), (1, 4), (5, 1), and (5, 4)?

A. 12
B. 15
C. 20
D. 10

Solution

Length = 5 - 1 = 4, Width = 4 - 1 = 3. Area = length * width = 4 * 3 = 12.

Correct Answer: A — 12

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Q. What is the circumference of a circle with a radius of 5?

A. 10π
B. 15π
C. 20π
D. 25π

Solution

Circumference = 2πr = 2π(5) = 10π.

Correct Answer: A — 10π

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Q. What is the equation of a line with a slope of 2 that passes through the point (1, 3)?

A. y = 2x + 1
B. y = 2x + 3
C. y = 2x - 1
D. y = 2x - 3

Solution

Using point-slope form: y - y1 = m(x - x1) => y - 3 = 2(x - 1) => y = 2x + 1.

Correct Answer: B — y = 2x + 3

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Q. What is the equation of the line that is perpendicular to y = 2x + 1 and passes through the point (1, 2)?

A. y = -1/2x + 5/2
B. y = 1/2x + 3/2
C. y = -2x + 4
D. y = 2x - 1

Solution

The slope of the perpendicular line is -1/2. Using point-slope form: y - 2 = -1/2(x - 1) => y = -1/2x + 5/2.

Correct Answer: A — y = -1/2x + 5/2

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Q. What is the equation of the line that passes through the point (2, 3) with a slope of 2?

A. y = 2x + 1
B. y = 2x - 1
C. y = 2x + 3
D. y = 2x - 3

Solution

Using point-slope form: y - y1 = m(x - x1) => y - 3 = 2(x - 2) => y = 2x - 4 + 3 => y = 2x - 1.

Correct Answer: A — y = 2x + 1

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Q. What is the equation of the line that passes through the points (1, 2) and (3, 4)?

A. y = x + 1
B. y = 2x
C. y = 2x - 1
D. y = x + 2

Solution

Slope = (4 - 2) / (3 - 1) = 1. Using point-slope form: y - 2 = 1(x - 1) => y = x + 1.

Correct Answer: A — y = x + 1

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

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

Solution

Using the Pythagorean theorem: d = √(length² + width²) = √(6² + 8²) = √(36 + 64) = √100 = 10.

Correct Answer: A — 10

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Q. What is the midpoint of the line segment connecting the points (2, 3) and (8, 7)?

A. (5, 5)
B. (4, 5)
C. (6, 4)
D. (7, 3)

Solution

Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 8)/2, (3 + 7)/2) = (5, 5).

Correct Answer: A — (5, 5)

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Q. What is the perimeter of a triangle with sides of lengths 5, 12, and 13?

A. 30
B. 25
C. 20
D. 18

Solution

Perimeter = 5 + 12 + 13 = 30.

Correct Answer: A — 30

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Q. What is the slope of the line passing through the points (1, 2) and (4, 6)?

A. 1
B. 2
C. 0.5
D. 3

Solution

Slope formula: m = (y2 - y1) / (x2 - x1) = (6 - 2) / (4 - 1) = 4 / 3.

Correct Answer: B — 2

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Basic Geometric Concepts - Coordinate Geometry Applications

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