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USA - Geometry Triangle theorems

Medians, altitudes, perpendicular bisectors, angle bisectors

20 practice questions 2 video lessons Theory + worked examples

Medians, Altitudes, and Bisectors

California Geometry • Standard G-CO.10 • Triangle Theorems

Medians, Altitudes, and Bisectors is a topic in Triangle Theorems in the California Common Core State Standards. It is aligned to Standard G-CO.10, which requires students to prove theorems about triangles and the special segments and points of concurrency within them.

The special segments of a triangle — medians, altitudes, perpendicular bisectors, and angle bisectors — meet at the centroid, orthocenter, circumcenter, and incenter.

California Geometry › Triangle Theorems › Medians, Altitudes, and Bisectors  —  Standard G-CO.10

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Practice questions

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Watch 2 video(s)
  • Incenter, Circumcenter, Centroid, Orthocenter (Properties & Diagrams) Watch
  • Incenter, Circumcenter, Orthocenter & Centroid of a Triangle - Geometry Watch
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Theory

Each triangle has four families of special segments, each set meeting at a single point of concurrency:

  • Median — vertex to the midpoint of the opposite side; the three meet at the centroid (the balance point), which divides each median \(2{:}1\).
  • Altitude — vertex perpendicular to the opposite side; the three meet at the orthocenter.
  • Perpendicular bisector of each side — the three meet at the circumcenter, equidistant from the vertices.
  • Angle bisector of each angle — the three meet at the incenter, equidistant from the sides.
Don't confuse median and altitude: a median hits the midpoint, an altitude meets the side at a right angle. They coincide only in special triangles.
Median and altitude from a vertex A median goes to the midpoint of the opposite side; an altitude is perpendicular to it. They are generally different segments. median (to midpoint) altitude (⊥)
A median (to the midpoint) and an altitude (perpendicular) from one vertex.
Points of concurrency Points of concurrency Points of concurrency medians → centroid (2:1) altitudes → orthocenter perp. bisectors → circumcenter angle bisectors → incenter
The four points of concurrency.

The concurrency points:

\[\text{medians}\to\text{centroid},\quad \text{altitudes}\to\text{orthocenter}\]
\[\text{perp. bisectors}\to\text{circumcenter},\quad \text{angle bisectors}\to\text{incenter}\]
medians meet at the centroid, altitudes at the orthocenter, perpendicular bisectors at the circumcenter, angle bisectors at the incenter
Centroid ratio \(2{:}1\): the vertex-to-centroid piece is twice the centroid-to-midpoint piece.

How to identify a special segment

  1. Median: ends at a side's midpoint.
  2. Altitude: meets a side at a right angle.
  3. Perpendicular bisector: perpendicular through a side's midpoint.
  4. Angle bisector: splits a vertex angle in half.
Example 1 — Identify the segment
A segment from a vertex to the midpoint of the opposite side is called what?
Solution

A segment to the midpoint of the opposite side is a median.

it is a median
Example 2 — Centroid ratio
The centroid divides each median in a \(2:1\) ratio. A median is \(12\). Find the longer piece.
Solution

The longer piece (vertex to centroid) is \(\dfrac{2}{3}\) of the median.

\(\dfrac{2}{3}\times 12\)\(=\)\(8\)
the longer piece is 8
Example 3 — Circumcenter
The circumcenter is the intersection of the three what?
Solution

The circumcenter is where the three perpendicular bisectors meet; it is equidistant from the three vertices.

the perpendicular bisectors, meeting at the circumcenter
Example 4 — Incenter
Which special segments meet at the incenter, and what is it equidistant from?
Solution

The three angle bisectors meet at the incenter, which is equidistant from the three sides.

the angle bisectors, at the incenter, equidistant from the sides

Common pitfalls

Median \(\ne\) altitude. One goes to the midpoint, the other is perpendicular; they match only in isosceles/equilateral cases.
Centroid divides \(2{:}1\), not in half. The longer piece is toward the vertex.
Circumcenter uses perpendicular bisectors; incenter uses angle bisectors. Keep the pairings straight.

Frequently asked questions

What is a median of a triangle?

A segment from a vertex to the midpoint of the opposite side.

What is the difference between a median and an altitude?

A median ends at the midpoint of a side; an altitude meets a side at a right angle.

What is the centroid and its ratio?

The point where the medians meet; it divides each median in a \(2{:}1\) ratio, closer to the midpoint.

What meets at the circumcenter and the incenter?

The perpendicular bisectors meet at the circumcenter (equidistant from vertices); the angle bisectors meet at the incenter (equidistant from sides).

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