Basic Geometric Concepts for Competitive Exam Success

Basic Geometric Concepts - Coordinate Geometry Applications - Applications

Q. If a triangle has vertices at (1, 2), (4, 6), and (1, 6), what is its area?

A. 10
B. 12
C. 8
D. 6

Solution

Area = 0.5 * |(1(6-6) + 4(6-2) + 1(2-6))| = 0.5 * |0 + 16 - 4| = 6.

Correct Answer: B — 12

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Q. What is the area of a circle with a radius of 4?

A. 16π
B. 8π
C. 12π
D. 20π

Solution

Area = πr² = π(4)² = 16π.

Correct Answer: A — 16π

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Q. What is the area of a triangle with a base of 10 and a height of 5?

A. 25
B. 30
C. 20
D. 15

Solution

Area = 0.5 * base * height = 0.5 * 10 * 5 = 25.

Correct Answer: A — 25

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Q. What is the equation of a line parallel to y = 3x + 2 that passes through the point (1, 1)?

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

Solution

Parallel lines have the same slope. Using point-slope form: y - 1 = 3(x - 1) gives y = 3x - 2.

Correct Answer: B — y = 3x + 1

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Q. What is the equation of a line that passes through the points (1, 2) and (3, 6)?

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

Solution

Slope = (6-2)/(3-1) = 2. Using point-slope form: y - 2 = 2(x - 1) gives y = 2x.

Correct Answer: A — y = 2x

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Q. What is the equation of the line that passes through the point (2, 3) with a slope of 4?

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

Solution

Using point-slope form: y - y1 = m(x - x1) => y - 3 = 4(x - 2) => y = 4x - 8 + 3 => y = 4x - 5.

Correct Answer: C — y = 4x - 3

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

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

Solution

First, find the slope: m = (3 - 2) / (2 - 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 perimeter of a triangle with sides measuring 5, 12, and 13?

A. 30
B. 25
C. 20
D. 15

Solution

Perimeter = side1 + side2 + side3 = 5 + 12 + 13 = 30.

Correct Answer: A — 30

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Q. What is the radius of a circle with the equation (x - 2)² + (y + 3)² = 25?

A. 5
B. 10
C. 25
D. 20

Solution

Radius = √25 = 5.

Correct Answer: A — 5

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Q. What is the slope of the line passing through the points (1, 2) and (3, 8)?

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

Solution

Slope (m) = (y2 - y1) / (x2 - x1) = (8 - 2) / (3 - 1) = 6 / 2 = 3.

Correct Answer: B — 4

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Q. What is the slope of the line that passes through the points (2, 3) and (5, 11)?

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

Solution

Slope = (11-3)/(5-2) = 8/3.

Correct Answer: B — 3

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Basic Geometric Concepts - Coordinate Geometry Applications - Applications

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