Basic Geometric Concepts - Coordinate Geometry Applications

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Q1. What is the area of a rectangle with vertices at (1, 1), (1, 4), (5, 1), and (5, 4)?

Solution:

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

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Q2. What is the equation of a line with a slope of 2 that passes through the point (1, 3)?

Solution:

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

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Q3. If a line has a slope of -3 and passes through the point (2, 5), what is its equation in slope-intercept form?

Solution:

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

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Q4. What is the equation of the line that passes through the point (2, 3) with a slope of 2?

Solution:

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

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Q5. What is the length of the diagonal of a rectangle with length 6 and width 8?

Solution:

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

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Q6. If a circle has a radius of 5, what is its area?

Solution:

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

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Q7. What is the equation of the line that is perpendicular to y = 2x + 1 and passes through the point (1, 2)?

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.

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Q8. If a line has a slope of -2 and passes through the point (3, 5), what is its y-intercept?

Solution:

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

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Q9. What is the circumference of a circle with a radius of 5?

Solution:

Circumference = 2πr = 2π(5) = 10π.

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Q10. What is the midpoint of the line segment connecting the points (2, 3) and (8, 7)?

Solution:

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