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Coordinate Geometry (Distance, Slope, Area)

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Introduction

Coordinate geometry connects algebra and geometry by placing points on the x-y plane. Skills like finding the distance between points, the slope of a line, and the area of a polygon (triangle) using coordinates are frequently tested in aptitude exams.

This pattern helps convert geometric questions into simple algebraic calculations using coordinate formulas.

Pattern: Coordinate Geometry (Distance, Slope, Area)

Pattern: Coordinate Geometry (Distance, Slope, Area)

Key concept: Use coordinate formulas to compute distances, slopes, midpoints and polygon areas directly from point coordinates.

Important formulas:
• Distance between (x₁, y₁) and (x₂, y₂): √[(x₂ - x₁)² + (y₂ - y₁)²].
• Slope of line through (x₁, y₁) and (x₂, y₂): (y₂ - y₁) / (x₂ - x₁) (vertical line → undefined).
• Midpoint of segment joining (x₁, y₁) and (x₂, y₂): ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |.

Step-by-Step Example

Question

Given points A(-1, 2), B(3, -2) and C(5, 4):
(a) Find the distance AB.
(b) Find the slope of BC.
(c) Find the area of triangle ABC.

Solution

  1. Step 1: Distance AB using distance formula.

    AB = √[(x₂ - x₁)² + (y₂ - y₁)²] with A(-1,2), B(3,-2).
    Compute differences: x₂ - x₁ = 3 - (-1) = 4; y₂ - y₁ = -2 - 2 = -4.
    AB = √(4² + (-4)²) = √(16 + 16) = √32 = 4√2.

  2. Step 2: Slope of BC using slope formula.

    B(3,-2), C(5,4). Slope m = (y₂ - y₁)/(x₂ - x₁) = (4 - (-2)) / (5 - 3) = 6/2 = 3.

  3. Step 3: Area of triangle ABC using determinant formula.

    Area = ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |.
    Substitute A(-1,2), B(3,-2), C(5,4):
    = ½ | (-1)((-2) - 4) + 3(4 - 2) + 5(2 - (-2)) |.
    Simplify inside: (-1)(-6) + 3(2) + 5(4) = 6 + 6 + 20 = 32.
    Area = ½ × |32| = 16 square units.

  4. Final Answers:

    (a) AB = 4√2.
    (b) Slope of BC = 3.
    (c) Area of ΔABC = 16.

  5. Quick Check:

    • Distance differences were equal magnitude → AB diagonal of a 4×4 right triangle → 4√2 makes sense.
    • Slope 3 is integer and positive as C is above B and to the right.
    • Area integer 16 is consistent with determinant arithmetic above ✅

Quick Variations

1. Find equation of line through two points (use point-slope form).

2. Check perpendicularity: slopes m₁·m₂ = -1.

3. Find midpoint and use as center for circles or reflections.

4. Area of polygon with >3 vertices: use shoelace formula (extension of triangle determinant).

Trick to Always Use

  • Step 1 → Always compute coordinate differences first (x₂ - x₁, y₂ - y₁) to avoid sign mistakes.
  • Step 2 → For slopes, simplify fraction early; watch for vertical (undefined) or horizontal (0) cases.
  • Step 3 → For area, use the determinant form (or shoelace) and keep track of sign; take absolute value at the end.

Summary

Coordinate geometry problems reduce to straightforward algebra if you remember key formulas:

  • Distance: √[(x₂ - x₁)² + (y₂ - y₁)²].
  • Slope: (y₂ - y₁)/(x₂ - x₁) (vertical → undefined).
  • Area of triangle: ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |.
  • Compute differences first, simplify fractions early, and always do a quick sanity check.

Practice

(1/5)
1. Find the distance between the points A(2, 3) and B(7, 11).
easy
A. 8.9 units
B. 9.2 units
C. 9.4 units
D. 9.8 units

Solution

  1. Step 1: Use distance formula.

    Distance = √[(x₂ - x₁)² + (y₂ - y₁)²].

  2. Step 2: Substitute values.

    = √[(7 - 2)² + (11 - 3)²] = √[5² + 8²] = √(25 + 64) = √89.

  3. Step 3: Simplify numeric value.

    √89 ≈ 9.43.

  4. Final Answer:

    Distance = 9.4 units → Option C.

  5. Quick Check:

    Square differences 25 + 64 = 89; √89 ≈ 9.43 ✅
Hint: Subtract coordinates, square differences, add, then take square root and round neatly.
Common Mistakes: Leaving root form when numeric value is expected or rounding incorrectly.
2. Find the slope of the line joining points P(1, 2) and Q(4, 8).
easy
A. 2
B. 1.5
C. 3
D. 4

Solution

  1. Step 1: Use slope formula.

    Slope m = (y₂ - y₁)/(x₂ - x₁).

  2. Step 2: Substitute values.

    m = (8 - 2)/(4 - 1) = 6/3.

  3. Step 3: Simplify.

    m = 2.

  4. Final Answer:

    Slope = 2 → Option A.

  5. Quick Check:

    For every 3 units in x, y rises by 6 → slope 2 ✅
Hint: Subtract y’s over x’s - rise over run.
Common Mistakes: Reversing x and y or using wrong order of subtraction.
3. Find the midpoint of the line joining points A(-4, 6) and B(8, 2).
easy
A. (2, 4)
B. (3, 3)
C. (1, 5)
D. (4, 4)

Solution

  1. Step 1: Use midpoint formula.

    Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

  2. Step 2: Substitute values.

    = ((-4 + 8)/2, (6 + 2)/2) = (4/2, 8/2).

  3. Step 3: Simplify.

    Midpoint = (2, 4).

  4. Final Answer:

    Midpoint = (2, 4) → Option A.

  5. Quick Check:

    Average of x’s and y’s gives midpoint ✅
Hint: Average x’s and y’s separately to get midpoint.
Common Mistakes: Adding without dividing by 2.
4. Find the area of a triangle with vertices A(1, 2), B(4, 6), and C(6, 2).
medium
A. 6 sq. units
B. 8 sq. units
C. 9 sq. units
D. 10 sq. units

Solution

  1. Step 1: Use area (determinant) formula.

    Area = ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |.

  2. Step 2: Substitute values.

    = ½ | 1(6 - 2) + 4(2 - 2) + 6(2 - 6) |.

  3. Step 3: Simplify.

    = ½ | 1×4 + 0 + 6×(-4) | = ½ |4 - 24| = ½ × 20 = 10 sq. units.

  4. Final Answer:

    Area = 10 sq. units → Option D.

  5. Quick Check:

    Determinant yields 20 in absolute value; half gives 10 ✅
Hint: Use determinant (shoelace) formula and take absolute value at the end.
Common Mistakes: Forgetting to multiply by ½ or to take absolute value.
5. The coordinates of two points are A(2, -3) and B(-4, 5). Find the slope of line AB.
medium
A. -4/3
B. 4/3
C. -3/4
D. 3/4

Solution

  1. Step 1: Apply slope formula.

    Slope = (y₂ - y₁)/(x₂ - x₁).

  2. Step 2: Substitute values.

    = (5 - (-3)) / (-4 - 2) = 8 / (-6) = -4/3.

  3. Final Answer:

    Slope = -4/3 → Option A.

  4. Quick Check:

    Negative slope as x increases, y decreases ✅
Hint: Subtract y’s over x’s; keep signs careful.
Common Mistakes: Subtracting in wrong order or missing negative signs.