# 2.9 – Medians and Altitudes

## Objectives

• Explore the medians and altitudes of triangles.
• Discover the centroids, orthocenters, incenters, and circumcenters of triangles.

## Key Terms

 Altitude of a Triangle – The line segment from a vertex of a triangle that is perpendicular to the opposite side. The altitude is sometimes OUTSIDE of the triangle The altitude is sometimes ON the triangle’s side (right triangles)   Median of a triangle – A line or segment joining a vertex of a triangle to the midpoint of the opposite side. In the example below, the median passes through vertex A and bisects side $\overline{BC}$. Orthocenter – The point at which the three altitudes of a triangle intersect. The othercenter is found (the purple dot in the image below): Inside acute triangles Outside obtuse triangles On the vertex of a the right angle for right triangles   Centroid of a triangle – The point at which the three medians of a triangle intersect. The centroid of any triangle is ALWAYS INSIDE the triangle How to Find the Centroid of a Triangle: Step 1: Find the midpoint of one side. Step 2: Draw a segment connecting the midpoint to its opposite vertex. Step 3: Repeat steps 1 and 2 for the other two sides. Step 4: The point where the three segments intersect is the centroid

## Notes

Altitudes and Medians
• Altitudes form orthocenters
• Medians form centroids

• Isosceles Triangles have dividing lines that are both: medians and altitudes