MathematicsChapter 10: Practical Geometry
Centres of A Triangle
A median joins a vertex to the midpoint of the opposite side.
Learning objectives
- 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.
What this grid covers
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.
Where / symbols
- 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 — 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).
Four centres, four constructions, four jobs:
| Centre | Made from | Where it sits |
|---|---|---|
| 1. in-centre | angle bisectors | always inside; centre of incircle |
| 2. orthocentre | altitudes | inside acute; at right vertex if right; outside obtuse |
| 3. circumcentre | perpendicular (right) bisectors | inside acute; midpoint of hypotenuse if right; outside obtuse |
| 4. centroid | medians (vertex to midpoint) | always inside; divides each median 2:1 |
Explanation
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.
In simple words
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).
Key points
- median
- midpoint
- centroid
- 2:1
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Centres of A Triangle
Sticky Notes · Quick Revision
WHAT
- A median joins a vertex to the midpoint of the opposite side.
Key Points
- A median joins a vertex to the midpoint of the opposite side.
- Four centres, four constructions, four jobs:
- in-centre: angle bisectors.
- orthocentre: altitudes.
HOW
- 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.
Examples
- in-centre: angle bisectors
- orthocentre: altitudes
- circumcentre: perpendicular (right) bisectors
- centroid: medians (vertex to midpoint)
Common Mistakes
- Correction: Median is vertex to midpoint, not perpendicular unless the triangle is isosceles.
- Correction: Centroid is always inside and splits each median 2:1.
- Correction: In-centre is also always inside.
Exam Tip
- 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).
60-second revision
- A median joins a vertex to the midpoint of the opposite side.
- Four centres, four constructions, four jobs:
- in-centre: angle bisectors.
- orthocentre: altitudes.
- 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.
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