Coordinate Geometry

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Q1. If the coordinates of point C are (5, 5) and the slope of the line is 1, what is the equation of the line? (2021) 2021

Solution:

Using point-slope form: y - 5 = 1(x - 5) => y = x.

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Q2. What is the area of a rectangle with vertices at (1, 1), (1, 5), (4, 1), and (4, 5)? (2021)

Solution:

Area = length * width = (4-1) * (5-1) = 3 * 4 = 12.

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Q3. 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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Q4. 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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Q5. What is the slope of the line perpendicular to the line y = 3x + 1? (2023)

Solution:

Slope of perpendicular line = -1/(slope of original line) = -1/3.

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Q6. If the coordinates of point C are (5, 5) and point D are (5, 10), what is the length of CD? (2021)

Solution:

Length CD = |10 - 5| = 5.

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Q7. 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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Q8. 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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Q9. Find the coordinates of the centroid of the triangle with vertices at (0, 0), (6, 0), and (0, 8).

Solution:

Centroid = ((0+6+0)/3, (0+0+8)/3) = (2, 2).

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Q10. What is the y-intercept of the line 2x - 3y + 6 = 0? (2023)

Solution:

Rearranging gives y = (2/3)x + 2; y-intercept = 2