Coordinate Geometry - Quick Notes & Formula Sheet
Class 10 CBSE Mathematics · Chapter 7 · Complete Revision Guide (2026-27)
2026-27 Syllabus Update - Read This First
The Area of a Triangle formula using coordinates has been completely removed from the CBSE 2026-27 board exam. This is a significant change: it also means the common shortcut of "checking collinearity by setting the area formula to zero" is no longer a valid board-exam method. For 2026-27, collinearity must be tested using the distance formula (checking if the sum of two smaller distances equals the longest distance) or the section formula (checking if the division ratio is consistent). This chapter now revolves entirely around three formulas: Distance, Section, and Midpoint.
1. Distance Formula
The distance between two points A(x₁,y₁) and B(x₂,y₂) on the coordinate plane.
AB = √[(x₂-x₁)² + (y₂-y₁)²]
Trap: Sign errors when subtracting coordinates
Students often make sign mistakes when one or both coordinates are negative. Always substitute carefully with brackets: (x₂-x₁) means take the SECOND point's x-coordinate minus the FIRST point's x-coordinate, consistently for both points - don't mix the order between x and y.
2. Section Formula (Internal Division)
The coordinates of a point P(x,y) that divides the line segment joining A(x₁,y₁) and B(x₂,y₂) internally in the ratio m:n.
P(x,y) = [ (mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n) ]
Trap: Swapping which point gets multiplied by which ratio part
A very common error: multiplying x₁ by m and x₂ by n (the reverse of correct). Remember: the ratio part CLOSER to a point gets multiplied by the OTHER point's coordinate - m (the part nearer B) multiplies x₂, and n (the part nearer A) multiplies x₁.
3. Midpoint Formula (Special Case of Section Formula)
When a point divides a segment in ratio 1:1 (exactly at the middle), the section formula simplifies directly.
Midpoint = [ (x₁+x₂)/2, (y₁+y₂)/2 ]
Quick check tip
The midpoint formula is simply the section formula with m=n=1 - you don't need to memorize it separately if you know the section formula well, but it's worth knowing by heart since it appears so frequently.
4. Testing Collinearity (Without the Area Formula)
Three points A, B, C are collinear (lie on the same straight line) if one of the following holds:
| Method | Condition |
|---|---|
| Distance Formula method | AB + BC = AC (where C is the point between A and the farthest point), i.e., the sum of the two smaller distances equals the largest distance |
| Section Formula method | If B divides AC in some ratio, that same ratio must be consistent when calculated from both the x-coordinates and y-coordinates separately |
Trap: Using the deleted area=0 method out of habit
Many older guides and PDFs still teach "if area of triangle formed by 3 points = 0, they are collinear." This is mathematically still true, but the formula itself is no longer in the syllabus, so board exams won't expect you to use it - rely on the distance-sum method instead.
5. Common Problem Patterns
| Problem Type | Approach |
|---|---|
| Find a missing coordinate given a ratio | Use the section formula, setting up two equations (one for x, one for y) and solving |
| Check if a quadrilateral is a parallelogram | Show that both diagonals have the same midpoint (diagonals of a parallelogram bisect each other) |
| Check if a triangle is isosceles/equilateral/right-angled | Compute all three side lengths using the distance formula, then compare (isosceles: 2 equal sides; right-angled: satisfies Pythagoras) |
| Find points that trisect a line segment | Apply the section formula twice, using ratios 1:2 and 2:1 from the same starting point |
Quick check tip
For "is this triangle right-angled" questions, compute all three squared distances first (skip the square root until the end) - this makes it much easier to directly check the Pythagoras relationship without rounding errors from early square roots.
6. Full Recap - Every Trap in One Place
| Trap | Correct Understanding |
|---|---|
| Sign errors when one/both coordinates are negative | Always substitute with brackets and simplify step by step |
| Swapping m and n in the section formula | m (ratio part near B) multiplies x₂/y₂; n (part near A) multiplies x₁/y₁ |
| Using the area=0 method for collinearity | Deleted for 2026-27 - use the distance-sum method or section-formula ratio consistency instead |
| Taking square roots too early in multi-step problems | Work with squared distances as long as possible to avoid rounding/decimal errors |
Frequently Asked Questions
What is the distance formula?
The distance between (x₁,y₁) and (x₂,y₂) is √[(x₂-x₁)²+(y₂-y₁)²].
What is the section formula?
It gives the coordinates of a point dividing a segment internally in a ratio m:n. The midpoint formula is the special case where the ratio is 1:1.
Is the area of a triangle formula included in CBSE 2026-27?
No - it's been fully removed. Only the distance formula and section formula (with midpoint as a special case) remain examinable in this chapter.
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