Lesson 5.1: Medians, Altitudes, Angle Bisectors

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

This lesson introduces three important triangle segments: medians, altitudes, and angle bisectors. Each connects a vertex to the opposite side in a different way and creates important triangle centers.

Learning Objectives

• Define triangle medians, altitudes, and angle bisectors.
• Identify each segment in geometric diagrams.
• Understand which triangle center each produces.
• Apply properties such as midpoint division and perpendicular distance.

Topics Covered

• Median and centroid
• Altitude and orthocenter
• Angle bisector and incenter
• Concurrency of triangle segments

Key Terms

• Median: Segment from a vertex to the midpoint of the opposite side.
• Centroid: Intersection of the three medians.
• Altitude: Perpendicular segment from a vertex to the opposite side.
• Orthocenter: Intersection of the three altitudes.
• Angle Bisector: Segment dividing an angle into two equal angles.
• Incenter: Intersection of the three angle bisectors.

Lesson Examples

Example 1: Finding a Median Ratio

The centroid divides each median in a 2:1 ratio from vertex to midpoint.

If full median = 12, then

\[ \text{vertex to centroid}=\frac{2}{3}(12)=8 \]

Example 2: Identifying an Altitude

1. Start at a triangle vertex.
2. Drop a perpendicular to the opposite side.
3. The perpendicular segment is the altitude.

Example 3: Angle Bisector Property

If an angle is \(60^\circ\), the bisector creates two angles:

\[ 30^\circ \text{ and } 30^\circ \]

Practice Problems

1. A triangle has median length 15. How far is the centroid from the vertex?
2. Draw a triangle and construct all three altitudes.
3. Bisect a \(50^\circ\) angle. What are the new angles?
4. Which segment always meets the opposite side at a right angle?

Tips & Common Mistakes

• Medians go to midpoints — not perpendicular unless the triangle is special.
• Altitudes must form 90° angles.
• Angle bisectors divide angles, not sides.

Summary

Medians, altitudes, and angle bisectors are three core triangle constructions. Each produces a special center and plays a major role in geometric proofs and constructions.

Challenge Problems

1. If the centroid divides a median and the longer section is 10, what is the full median?
2. Can an altitude lie outside the triangle? Draw an example.
3. Explain why angle bisectors always meet inside the triangle.