Coordinate Geometry: Where Algebra Meets the Map

Learn the distance formula, section formula, and midpoint formula to solve Class 10 Coordinate Geometry problems on the x-y plane with confidence.

By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026

Part of Class 10 (CBSE) prep

🧠 A city with street numbers

Think of a planned city like Chandigarh, where every location is given by a sector number across and a number up. Tell an autorickshaw driver '3 blocks east, 4 blocks north' and they find you exactly. Coordinate geometry does the same thing for points on a graph: every point gets an address (x, y), and once points have addresses, we can measure distances and find meeting spots using simple formulas.

The plane is split by two number lines: the horizontal x-axis and the vertical y-axis, meeting at the origin (0, 0). A point's position is written as (x, y), where x is how far right or left it sits and y is how far up or down. With this system, geometry problems turn into algebra you can compute.

The three formulas you must know

x y I (+,+)II (−,+)III (−,−)IV (+,−) P (3, 2) 123 12
Every point is an (x, y) pair; the axes split the plane into four quadrants.
Distance between P(x1, y1) and Q(x2, y2): PQ = square root of [(x2 - x1)^2 + (y2 - y1)^2].
Section formula - point dividing the join of (x1, y1) and (x2, y2) in ratio m:n is ( (m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n) ).
Midpoint of (x1, y1) and (x2, y2): ( (x1 + x2)/2, (y1 + y2)/2 ). This is the section formula with ratio 1:1.
💡 The distance from the origin (0, 0) to any point (x, y) is simply the square root of (x^2 + y^2). It is just the distance formula with one point set to the origin - a handy shortcut in exams.

📝 Distance between two points

Find the distance between A(2, 3) and B(5, 7).

Distance formula: AB = square root of [(x2 - x1)^2 + (y2 - y1)^2].

Substitute: AB = square root of [(5 - 2)^2 + (7 - 3)^2].

= square root of [3^2 + 4^2] = square root of [9 + 16] = square root of 25.

So AB = 5 units.

📝 Using the section formula

Find the point that divides the line joining A(-1, 2) and B(4, 7) in the ratio 2:3.

Here (x1, y1) = (-1, 2), (x2, y2) = (4, 7), m = 2, n = 3.

x-coordinate = (m x2 + n x1)/(m + n) = (2 x 4 + 3 x (-1))/(2 + 3) = (8 - 3)/5 = 5/5 = 1.

y-coordinate = (m y2 + n y1)/(m + n) = (2 x 7 + 3 x 2)/(2 + 3) = (14 + 6)/5 = 20/5 = 4.

So the required point is (1, 4).

⚠️ Order matters in the section formula. The ratio m:n means the point is m parts from the first point and n parts from the second. Mixing up which point is (x1, y1) and which is (x2, y2) is the most common exam error - label them before you substitute.
TaskFormula to use
Length of a segmentDistance formula
Point dividing a segment in ratio m:nSection formula
Exact middle of a segmentMidpoint formula
Check if 3 points are collinearDistance formula (sum of two = third) or section formula

Quick revision

  • ✓Every point has an address (x, y): x is horizontal, y is vertical.
  • ✓Distance formula comes straight from the Pythagoras theorem on the plane.
  • ✓Midpoint is the section formula with ratio 1:1.
  • ✓Distance from origin to (x, y) is square root of (x^2 + y^2).
  • ✓Note: the area-of-triangle formula was moved out of the current NCERT Class 10 chapter, so focus on distance, section, and midpoint.

⚡ Quick check

What is the midpoint of the line segment joining the points A(-2, 4) and B(6, -2)?

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