Special Segments in Triangles
A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.
Four classical segments live in every triangle. A MEDIAN goes from a vertex to the midpoint of the opposite side. An ALTITUDE drops perpendicularly from a vertex to the opposite side (or its extension). A PERPENDICULAR BISECTOR is perpendicular to a side at its midpoint (it usually doesn't pass through a vertex). An ANGLE BISECTOR splits an angle in half and starts at a vertex. Three of each kind always meet at a concurrency point (centroid, orthocenter, circumcenter, incenter respectively).
What you'll learn
- Identify medians, altitudes, perpendicular bisectors, and angle bisectors in a triangle
- Know that the centroid divides each median in a 2:1 ratio (vertex side : midpoint side)
- Recognize each segment's defining property
Worked example
Problem. A median connects a vertex to the midpoint of the opposite side. The three medians meet at the centroid, which divides each median in a ___ ratio measured from the vertex.
- The centroid is always 2/3 of the way from each vertex to the opposite side's midpoint.
- So vertex-to-centroid : centroid-to-midpoint = 2 : 1.
- Connect the result back to Special Segments in Triangles so the geometric relationship is explicit.
- Check the result against the original diagram, units, or proof statement.
Answer: 2:1
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. A median goes from a vertex to:
Choices: The midpoint of the opposite side · The foot of the altitude · The opposite vertex
Show solution
- Definition of a median.
Answer: The midpoint of the opposite side
2. An altitude is:
Choices: Perpendicular from a vertex to the opposite side · From midpoint to midpoint
Show solution
- Definition of an altitude.
Answer: Perpendicular from a vertex to the opposite side
3. Three medians meet at the:
Choices: Centroid · Incenter · Circumcenter · Orthocenter
Show solution
- The medians concur at the centroid.
Answer: Centroid
4. Three altitudes meet at the:
Choices: Centroid · Incenter · Circumcenter · Orthocenter
Show solution
- Altitudes concur at the orthocenter.
Answer: Orthocenter
5. Three perpendicular bisectors meet at the:
Choices: Centroid · Incenter · Circumcenter · Orthocenter
Show solution
- Perpendicular bisectors concur at the circumcenter.
Answer: Circumcenter
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