Chapter 5 of 9

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Tamil Nadu (TNBSE)Class 9 Mathematics★ 6-10 marks in the TN Class 9 annual exam through MCQs, distance, mid-point and centroid problems

Coordinate Geometry

Samacheer Kalvi (TN State Board) Class 9 Maths Chapter 5, Coordinate Geometry. The Cartesian coordinate system, the distance formula, the mid-point and section formulas, points of trisection, and the centroid of a triangle. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 20 min readEasy difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Plot points and identify quadrants
2Apply the distance formula
3Use the mid-point and section formulas
4Find points of trisection
5Calculate the centroid of a triangle

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Why this chapter matters

Coordinate Geometry links algebra with the plane. The distance, mid-point, section and centroid formulas give reliable, formula-driven marks in the TN Class 9 exam and prepare students for Class 10 coordinate geometry.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Class 8 — Introduction to coordinates

Plotting points on the plane.

Coordinate Geometry — Class 9 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 9 Mathematics, Chapter 5. Locating points and measuring distances in the plane.

1. About this chapter

This chapter covers the Cartesian system, the distance formula, the mid-point and section formulas, points of trisection, and the centroid.

2. The Cartesian system

Two perpendicular axes (x and y) meet at the origin (0, 0), dividing the plane into four quadrants.
A point is written as an ordered pair (x, y).

3. Distance and mid-point

Distance formula: between A(x₁, y₁) and B(x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²].
Mid-point formula: mid-point of AB = ((x₁ + x₂)/2, (y₁ + y₂)/2).

4. Section formula and centroid

Section formula (point dividing AB in ratio m : n internally): ((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)).
Points of trisection divide a segment into three equal parts (ratios 1 : 2 and 2 : 1).
Centroid of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).

5. Worked examples

Example 1. Find the distance between (1, 2) and (4, 6). d = √[(4 − 1)² + (6 − 2)²] = √(9 + 16) = √25 = 5.

Example 2. Find the mid-point of (2, 3) and (6, 7). = ((2 + 6)/2, (3 + 7)/2) = (4, 5).

Example 3. Find the centroid of the triangle with vertices (0,0), (6,0), (0,9). = ((0+6+0)/3, (0+0+9)/3) = (2, 3).

6. Common mistakes

Mistake: Forgetting to square inside the distance formula. Fix: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
Mistake: Swapping m and n in the section formula. Fix: The ratio m : n multiplies the far then near coordinate carefully — write the formula before substituting.
Mistake: Dividing the centroid by 2 instead of 3. Fix: The centroid uses the average of three vertices (÷ 3).

7. Practice (book-back style)

Write the distance formula.
Find the distance between (0, 0) and (3, 4).
Find the mid-point of (−2, 5) and (4, 1).
Find the centroid of (1, 2), (3, 4), (5, 0).
State the section formula.

8. Answer key

d = √[(x₂ − x₁)² + (y₂ − y₁)²].
√(9 + 16) = 5.
((−2 + 4)/2, (5 + 1)/2) = (1, 3).
((1+3+5)/3, (2+4+0)/3) = (3, 2).
((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)).

9. Quick revision

Chapter 5 · distance, mid-point, section, centroid.
Distance d = √[(x₂−x₁)² + (y₂−y₁)²].
Mid-point = ((x₁+x₂)/2, (y₁+y₂)/2).
Section (m:n) = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
Centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Distance formula

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Between two points.

Mid-point formula

((x₁ + x₂)/2, (y₁ + y₂)/2)

Average of the coordinates.

Section formula

((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n))

Internal division m : n.

Centroid

((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)

Average of three vertices.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting to square inside the distance formula

✓ d = √[(x₂ − x₁)² + (y₂ − y₁)²].

WATCH OUT

✗ Swapping m and n in the section formula

✓ Write the formula before substituting and keep the ratio order.

WATCH OUT

✗ Dividing the centroid by 2 instead of 3

✓ The centroid averages three vertices (÷ 3).

NCERT exercises

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Multiple choice questions

MCQs on quadrants and formulas.

Questions

Distance / mid-point

Distance, mid-point and section problems.

Questions

Centroid / trisection

Centroid and points-of-trisection problems.

Questions

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Coordinate Geometry?

6 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

6 questions~4 min worth ~10 marks in Tamil Nadu (TNBSE) exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Chapter 5 of Samacheer Kalvi Class 9 Mathematics.
•Distance d = √[(x₂−x₁)² + (y₂−y₁)²].
•Mid-point = ((x₁+x₂)/2, (y₁+y₂)/2).
•Section (m:n) = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
•Centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
•Mid-point is the section formula with m = n = 1.

Tamil Nadu (TNBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-10 marks across MCQ, distance and centroid problems

Question type	Marks each	Typical count	What it tests
MCQ	1	1-2	Quadrants and formulas
Short Answer	2-3	1-2	Distance, mid-point, section
Centroid / Trisection	2-3	1	Centroid and trisection points

Prep strategy

Memorise the four formulas
Practise distance and mid-point sums
Use the section formula carefully
Average three vertices for the centroid

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Mapping

Distances and mid-points locate places on coordinate maps.

Computer graphics

Points and centroids position and balance shapes.

Navigation

Coordinates describe routes and positions.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Write the formula before substituting

Square the differences in the distance formula

Keep the ratio order in the section formula

Divide by 3 for the centroid

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

STRETCH

Show that the diagonals of a parallelogram bisect each other using mid-points.

STRETCH

Find a point dividing a segment externally in a given ratio.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

TN Class 9 Annual ExamHigh

Foundation / NTSE MathematicsMedium

School unit testsHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

When the ratio is 1 : 1, the section formula reduces to the average of the coordinates, which is exactly the mid-point.

It is the point where the three medians meet — the triangle's balancing point — found by averaging the vertices' coordinates.

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Tamil Nadu (TNBSE) Class 9 mock tests

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