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Centres of a Triangle
A median joins a vertex to the midpoint of the opposite side. The three medians are concurrent at the centroid. The centroid divides each median in the ratio 2:1, with the longer part towards the vertex. Together with the in-centre, orthocentre, and circumcentre you now have four named centres. Only the in-centre and the centroid are always inside every triangle.
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Centres of a Triangle
What is it?
A median joins a vertex to the midpoint of the opposite side. The three medians are concurrent at the centroid. The centroid divides each median in the ratio 2:1, with the longer part towards the vertex. Together with the in-centre, orthocentre, and circumcentre you now have four named centres. Only the in-centre and the centroid are always inside every triangle.
In simple words
- A median joins a vertex to the midpoint of the opposite side.
- Four centres, four constructions, four jobs:
- in-centre: angle bisectors.
- orthocentre: altitudes.
Important terms
median: A median joins a vertex to the midpoint of the opposite side.
midpoint: A median joins a vertex to the midpoint of the opposite side.
centroid: The three medians are concurrent at the centroid.
2:1: The centroid divides each median in the ratio 2:1, with the longer part towards the vertex.
What you should learn
- Explain centres of a triangle in clear mathematical language.
- Use the main rule, formula, test, or construction correctly.
- Work through a simple example and show each important step.
- Identify common mistakes and write an exam-ready conclusion.
Recommended for Class 9
Full school-level explanation
A median joins a vertex to the midpoint of the opposite side. The three medians are concurrent at the centroid. The centroid divides each median in the ratio 2:1, with the longer part towards the vertex. Together with the in-centre, orthocentre, and circumcentre you now have four named centres. Only the in-centre and the centroid are always inside every triangle.
WHAT — Core idea
A median joins a vertex to the midpoint of the opposite side. The three medians are concurrent at the centroid. The centroid divides each median in the ratio 2:1, with the longer part towards the vertex. Together with the in-centre, orthocentre, and circumcentre you now have four named centres. Only the in-centre and the centroid are always inside every triangle.
HOW — Rules, method, and examples
Four centres, four constructions, four jobs: in-centre: angle bisectors; always inside; centre of incircle. orthocentre: altitudes; inside acute; at right vertex if right; outside obtuse. circumcentre: perpendicular (right) bisectors; inside acute; midpoint of hypotenuse if right; outside obtuse. centroid: medians (vertex to midpoint); always inside; divides each median 2:1.
WHY — Meaning and connections
To draw a median, find the midpoint of a side (compass equal arcs from both ends, or a ruler), then join it to the opposite vertex. Repeat for a second median; they meet at the centroid, and the third median is forced through that point. The 2:1 split means vertex-to-centroid is twice centroid-to-midpoint. In an equilateral triangle all four centres coincide.
HOW — Answer in an exam
Median is vertex to midpoint, not perpendicular unless the triangle is isosceles. Centroid is always inside and splits each median 2:1. In-centre is also always inside. Orthocentre and circumcentre can leave the triangle. Right triangle: circumcentre at the midpoint of the hypotenuse (diameter).
Remember
- median
- midpoint
- centroid
- 2:1
- midpoint of a side
- twice
- coincide
- 2:1
How the ideas connect
To draw a median, find the midpoint of a side (compass equal arcs from both ends, or a ruler), then join it to the opposite vertex. Repeat for a second median; they meet at the centroid, and the third median is forced through that point. The 2:1 split means vertex-to-centroid is twice centroid-to-midpoint. In an equilateral triangle all four centres coincide.
Deeper understanding
Go beyond memorising Centres of a Triangle. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
To draw a median, find the midpoint of a side (compass equal arcs from both ends, or a ruler), then join it to the opposite vertex. Repeat for a second median; they meet at the centroid, and the third median is forced through that point. The 2:1 split means vertex-to-centroid is twice centroid-to-midpoint. In an equilateral triangle all four centres coincide.
Common mistakes
Exam focus
WHAT
Key Points
HOW
Examples
Exam focus
Quick revision
- median
- midpoint
- centroid
- 2:1
- median: A median joins a vertex to the midpoint of the opposite side.
- midpoint: A median joins a vertex to the midpoint of the opposite side.
Median is vertex to midpoint, not perpendicular unless the triangle is isosceles. Centroid is always inside and splits each median 2:1. In-centre is also always inside. Orthocentre and circumcentre can leave the triangle. Right triangle: circumcentre at the midpoint of the hypotenuse (diameter).
For long answers on Centres of a Triangle: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- median
- midpoint
- centroid
- 2:1
Important formulas
- median
- midpoint
- centroid
Common exam traps
- Correction: Median is vertex to midpoint, not perpendicular unless the triangle is isosceles.
- Correction: Centroid is always inside and splits each median 2:1.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Correction: Median is vertex to midpoint, not perpendicular unless the triangle is isosceles.
- Mistake
Correction: Centroid is always inside and splits each median 2:1.
- Mistake
Correction: In-centre is also always inside.
- Mistake
Correction: Orthocentre and circumcentre can leave the triangle.
- Mistake
Median is vertex to midpoint, not perpendicular unless the triangle is isosceles. Centroid is always inside and splits each median 2:1. In-centre is also always
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
median
midpoint
centroid
2:1
A median joins a vertex to the midpoint of the opposite side. The three medians are concurrent at the centroid. The centroid divides each median in the ratio 2:
- Definition
- A median joins a vertex to the midpoint of the opposite side. The three medians are concurrent at the centroid. The centroid divides each median in the ratio 2:
- Critical warning
- Correction: Median is vertex to midpoint, not perpendicular unless the triangle is isosceles.
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Exam Practice
Define
Q1. Define Centres of a Triangle.
Explain
Q2. Explain: WHAT — Core idea.
Example
Q3. Give one clear example of Centres of a Triangle and say why it fits.
Mistake
Q4. State one common mistake students make about Centres of a Triangle.
Exam note
Q5. Write a short exam note on Centres of a Triangle: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.