Coordinate Geometry

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Q1. What is the length of the diagonal of a rectangle with vertices at (0, 0), (4, 0), (4, 3), and (0, 3)? (2021)

Solution:

Length of diagonal = √[(4-0)² + (3-0)²] = √[16 + 9] = √25 = 5.

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Q2. What is the distance between the points (3, 4) and (7, 1)? (2021) 2021

Solution:

Distance = √[(7-3)² + (1-4)²] = √[4 + 9] = √13 ≈ 5.

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Q3. What is the length of the diagonal of a rectangle with vertices at (1, 1), (1, 4), (5, 1), and (5, 4)? (2021)

Solution:

Diagonal length = √[(5-1)² + (4-1)²] = √[16 + 9] = √25 = 5.

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Q4. What is the coordinates of the centroid of the triangle with vertices (0, 0), (6, 0), and (3, 6)? (2022)

Solution:

Centroid = ((0+6+3)/3, (0+0+6)/3) = (3, 2).

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Q5. What is the equation of the line passing through (0, 0) with a slope of 3? (2021)

Solution:

Equation of line: y = mx + c; here m = 3, c = 0 => y = 3x

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Q6. What is the equation of the line passing through (2, 3) with a slope of 2? (2021)

Solution:

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

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Q7. What is the distance from the point (1, 1) to the line x + y = 2? (2023)

Solution:

Distance = |1 + 1 - 2| / √(1² + 1²) = 0 / √2 = 0.

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Q8. Find the midpoint of the line segment joining the points (2, 3) and (4, 7). (2022) 2022

Solution:

Midpoint = ((2+4)/2, (3+7)/2) = (3, 5).

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Q9. What is the distance from the point (2, 3) to the line x + y = 5? (2020)

Solution:

Distance = |(1*2 + 1*3 - 5)| / √(1² + 1²) = |5 - 5| / √2 = 0/√2 = 0

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Q10. What is the distance from the point (1, 2) to the line x + y = 5? (2020)

Solution:

Distance = |1 + 2 - 5| / √(1² + 1²) = | -2 | / √2 = 2/√2 = √2.