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Grade 10/ Maths/ Coordinate Geometry
Chapter 7 · NCERT Maths 041

Coordinate Geometry

Give every point an address (x, y) and geometry becomes arithmetic — you can measure a distance or find a midpoint with a formula instead of a ruler. It's how GPS and games place everything.

📈 3 topics⏱ ~40 min📝 20-question quiz
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Distance, midpoint & section, live

Move points A and B. The distance AB, the midpoint (green), and a section point P that divides AB in your chosen ratio all update from their formulas.

A(x₁, y₁)   B(x₂, y₂)drag the sliders
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The three ideas in this chapter

Two number lines — the horizontal x-axis and the vertical y-axis — cross at the origin (0, 0). Any point is then named by an ordered pair (x, y): how far right/left, then how far up/down.

  • The axes split the plane into four quadrants (I, II, III, IV, anticlockwise from top-right).
  • On the x-axis, y = 0; on the y-axis, x = 0.
Watch out: order matters — (3, 5) is not the same point as (5, 3). Always read x first, then y.

The horizontal gap is (x₂ − x₁), the vertical gap is (y₂ − y₁). These two form the legs of a right triangle, so by Pythagoras the distance (the hypotenuse) is:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Worked example · distance between (1, 2) and (4, 6)
  1. x₂ − x₁ = 4 − 1 = 3; y₂ − y₁ = 6 − 2 = 4.
  2. d = √(3² + 4²) = √(9 + 16) = √25.
  3. d = 5 units.
Common mistake: forgetting to square, or adding the gaps directly. It's √(Δx² + Δy²), never Δx + Δy.

The point P that divides the segment from A(x₁, y₁) to B(x₂, y₂) in the ratio m : n is:

P = ( (mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n) )

The midpoint is just the special case m : n = 1 : 1:

M = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
Worked example · divide (1, 1)–(4, 7) in ratio 1 : 2
  1. m = 1, n = 2; A = (1, 1), B = (4, 7).
  2. x = (1·4 + 2·1)/(1+2) = (4 + 2)/3 = 2.
  3. y = (1·7 + 2·1)/3 = (7 + 2)/3 = 3 → P = (2, 3).
Common mistake: swapping m and n. The ratio m : n means m goes with the far point B (mx₂) and n with the near point A (nx₁). Keep the pairing straight.

Why this matters

Where you'll actually use this

Every map app, every video game, every delivery route turns the world into (x, y) coordinates and then uses exactly these two formulas to measure distances and place things.

Maps, GPS & "how far?"

Your phone stores places as coordinates. To tell you the straight-line distance between two pins, it runs the distance formula — √(Δx² + Δy²) — on their (x, y) values. Ride-hailing apps use it to find the nearest driver, and to estimate fares.

Distance formula
(−2, 3) (4, −3) midpoint = spawn between players

Games & computer graphics

Every character, bullet and pixel lives at an (x, y) (or x, y, z). Games use the midpoint/section formula to place objects between points, smooth camera moves, and the distance formula to check collisions and ranges — thousands of times per second.

Section & midpoint
🚚 Delivery routing

Logistics apps cluster nearby orders using coordinate distances to plan efficient routes.

✈️ Air-traffic spacing

Controllers keep aircraft safely apart by computing distances between their plotted positions.

🏗️ CAD & architecture

Design software places every wall and joint by coordinates and the section formula.

🔒 More real-world applications

Delivery routing, air-traffic spacing, CAD design and more — each explained with a diagram. Free to unlock.

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Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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Interactive explorer hand-built for trykarkedekho. Content from the rationalised NCERT Class 10 Maths syllabus (ncert.nic.in).

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