Coordinate Geometry — Class 10 Mathematics

"Geometry and algebra unite — a point becomes a pair of numbers, a line becomes an equation."

1. About the Chapter

Coordinate Geometry (or Cartesian Geometry) describes geometric shapes using algebra. Founded by René Descartes (1637) and earlier influenced by Indian mathematicians.

Key Idea

Every point in the plane is identified by (x, y) — ordered pair of coordinates.

What This Chapter Covers

• Recap of Cartesian plane
• Distance Formula
• Section Formula (internal division)
• Area of Triangle from coordinates
• Applications

2. The Cartesian Plane (Quick Review)

Axes

• x-axis: horizontal
• y-axis: vertical
• They meet at ORIGIN (0, 0)

Quadrants

• Q1: top-right, (+x, +y)
• Q2: top-left, (−x, +y)
• Q3: bottom-left, (−x, −y)
• Q4: bottom-right, (+x, −y)

Plot a Point

P(3, 5): go 3 units right, 5 units up. Q(−2, −4): go 2 units left, 4 units down.

3. Distance Formula

Statement

Distance between P(x₁, y₁) and Q(x₂, y₂):

PQ = √((x₂−x₁)² + (y₂−y₁)²)

Derivation

Form right triangle: horizontal leg = (x₂−x₁), vertical leg = (y₂−y₁). Apply Pythagoras.

Examples

Example 1: Find distance between A(2, 3) and B(5, 7).

• PQ = √((5−2)² + (7−3)²) = √(9 + 16) = √25 = 5 units

Example 2: Find distance from origin (0,0) to P(−3, 4).

• = √(9 + 16) = √25 = 5 units

Example 3: Verify (0,0), (3,4), (−3,4) form an isosceles triangle.

• d₁ from (0,0) to (3,4): 5
• d₂ from (0,0) to (−3,4): 5
• d₃ from (3,4) to (−3,4): 6
• Two sides equal (5 = 5), so isosceles ✓

4. Section Formula (Internal Division)

Statement

If P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) in ratio m:n (internally):

x = (mx₂ + nx₁) / (m + n) y = (my₂ + ny₁) / (m + n)

Midpoint (Special Case)

When P is the midpoint, m = n = 1: Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)

Examples

Example 1: Find point dividing line joining A(2, 3) and B(8, 9) in ratio 2:1.

• x = (2·8 + 1·2)/3 = 18/3 = 6
• y = (2·9 + 1·3)/3 = 21/3 = 7
• Point: (6, 7)

Example 2: Find midpoint of A(4, −3) and B(8, 5).

• ((4+8)/2, (−3+5)/2) = (6, 1)

5. Area of Triangle

Formula

For triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):

Area = (1/2) |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Note

• Use absolute value (area is always positive)
• If area = 0, points are COLLINEAR (on same line)

Examples

Example 1: Find area of triangle with vertices A(1, 2), B(4, 5), C(6, 1).

• Area = (1/2) |1(5−1) + 4(1−2) + 6(2−5)|
• = (1/2) |4 − 4 − 18|
• = (1/2)(18) = 9 sq units

Example 2: Check if A(2, 4), B(4, 6), C(6, 8) are collinear.

• Area = (1/2) |2(6−8) + 4(8−4) + 6(4−6)|
• = (1/2) |−4 + 16 − 12|
• = (1/2)(0) = 0
• YES, collinear.

6. Applications

Coordinate Geometry in Various Problems

Type 1: Equation of straight line

• Slope = (y₂ − y₁) / (x₂ − x₁)
• Line: y − y₁ = m(x − x₁)

Type 2: Parallel lines have EQUAL slopes Type 3: Perpendicular lines have slopes m₁ × m₂ = −1 Type 4: Find center of circle Type 5: Find centroid of triangle = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)

7. Worked Examples

Example 1: Equilateral Triangle

Show that points (0, 0), (3, √3), and (3, −√3) form an equilateral triangle.

• Side 1: from (0,0) to (3, √3) = √(9 + 3) = √12 = 2√3
• Side 2: from (0,0) to (3, −√3) = √(9 + 3) = 2√3
• Side 3: from (3, √3) to (3, −√3) = √(0 + 12) = 2√3
• All sides equal = 2√3 → EQUILATERAL ✓

Example 2: Find Coordinates

Find coordinates of points trisecting line joining A(2, 1) and B(5, 8).

• One trisecting point divides 1:2:
  • x = (1·5 + 2·2)/3 = 9/3 = 3
  • y = (1·8 + 2·1)/3 = 10/3
  • Point: (3, 10/3)
• Other trisecting point divides 2:1:
  • x = (2·5 + 1·2)/3 = 12/3 = 4
  • y = (2·8 + 1·1)/3 = 17/3
  • Point: (4, 17/3)

Example 3: Centroid

Find centroid of triangle with vertices (0, 0), (6, 0), (3, 9).

• Centroid = ((0+6+3)/3, (0+0+9)/3) = (3, 3)

Example 4: Circle Centre

A circle passes through (0, 0), (2, 0), and (0, 2). Find centre.

• Centre is equidistant from all three points.
• Let centre = (h, k). Distance to each = r.
• h² + k² = (h−2)² + k² → 4h = 4 → h = 1
• h² + k² = h² + (k−2)² → 4k = 4 → k = 1
• Centre: (1, 1)

8. Common Mistakes

1. Wrong coordinate order

   • Always (x, y) — first horizontal, then vertical.

2. Forgetting absolute value in area

   • Use |...| to ensure positive area.

3. Section formula in external division

   • For internal: (mx₂ + nx₁)/(m+n). For external (not in Class 10 syllabus normally): (mx₂ − nx₁)/(m−n).

4. Mixing up internal and external ratios

   • 'Internal' means point INSIDE the segment.

5. Wrong midpoint formula

   • Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2). Sum first, then divide.

9. Indian Heritage

While Cartesian coordinates were formalised by Descartes (1637), Indian mathematicians used:

• Bhaskara II (12th century): worked with curves and surfaces
• Vedic geometry: precise altar geometry
• Aryabhata (5th century): used coordinate-like thinking

Modern coordinate geometry combines Indian algebraic ideas with European geometric notation.

10. Conclusion

Coordinate Geometry bridges algebra and geometry:

• Points become numerical pairs
• Lines become equations
• Geometric problems solved algebraically

Master:

• Distance Formula
• Section Formula
• Area Formula

Foundation for:

• Class 11 Conic Sections
• Class 12 3D Geometry, Calculus
• Engineering, Physics
• Computer Graphics

The Cartesian Plane is the canvas on which all of mathematics is drawn.