Medians, Altitudes and Centroids
Special segments inside a triangle meet at famous points.
About this topic
A median joins a vertex to the midpoint of the opposite side, an altitude drops perpendicularly from a vertex to the opposite side, and the perpendicular bisectors and angle bisectors are other special segments. The three medians meet at the centroid, the triangle's balance point, which divides each median in a 2 : 1 ratio.
The altitudes meet at the orthocenter, the angle bisectors at the incenter, and the perpendicular bisectors at the circumcenter. These centers have uses in design and physics.
Worked example: a triangle has vertices (0, 0), (6, 0) and (3, 9). The centroid is the average of the coordinates: (3, 3). A median from (3, 9) to the midpoint (3, 0) has length 9, and the centroid is 6 units from the vertex and 3 from the base, the 2 : 1 split. A cardboard triangle balances on a pencil tip at that point.
Engineers use the centroid to find centres of mass for beams, boats and aircraft. A common confusion is mixing the four centres: medians give the centroid, altitudes the orthocenter, angle bisectors the incenter and perpendicular bisectors the circumcenter. In an equilateral triangle all of them coincide. Also remember that an altitude need not lie inside the triangle: in an obtuse triangle it can fall outside, and in a right triangle two altitudes are the legs.
Key ideas
- Medians meet at the centroid, the balance point
- Altitudes meet at the orthocenter
- Perpendicular bisectors meet at the circumcenter
Quick check
Use cases: where Medians, Altitudes and Centroids helps
- Homework: review the key idea behind Medians, Altitudes and Centroids before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Medians, Altitudes and Centroids without notes.
- Visual learning: play with the interactive lab to see how Medians, Altitudes and Centroids changes when you adjust its inputs.
- Teaching a friend: explain Medians, Altitudes and Centroids in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Medians, Altitudes and Centroids?
Special segments inside a triangle meet at famous points.
What will I learn in Medians, Altitudes and Centroids?
Medians meet at the centroid, the balance point. Altitudes meet at the orthocenter. Perpendicular bisectors meet at the circumcenter.
Is Medians, Altitudes and Centroids part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Medians, Altitudes and Centroids lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Medians, Altitudes and Centroids?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.