Geometry Notes Medians and Altitudes

Geometry Medians and Altitudes

Median of a Triangle A median of a triangle is a segment that joins the vertex of angle to the midpoint of the opposite side.

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Because there are three vertices, there are three possible medians.

One of the fascinating things about them is that no matter what shape the triangle, all three always intersect at a single point.

This point is called the centroid of the triangle.

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This intersection of three lines at a single point is called concurrent.

Two or more lines are said to be concurrent if they intersect in a single point.

There are some fascinating properties of the medians of a triangle: 1. The three medians always meet at a single point 2. Each median divides the triangle into two smaller triangles which have the same area 3. The centroid (point where they meet) is the center of gravity of the triangle.

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Centroid facts 1. The centroid is always inside the triangle 2. The centroid is exactly two-thirds the way along each median. Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.

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Altitude An altitude is a segment from the vertex of an angle that is perpendicular to the opposite side or to the line containing the opposite side. A triangle has three A altitudes.

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An interesting fact is that the three altitudes always pass through a common point called the orthocenter of the triangle.

The orthocenter is not always inside the triangle.

If the triangle is obtuse, it will be outside. To make this happen the altitude lines have to be extended so they cross.

Summary of triangle centers.

There are many types of triangle centers. Below are four of the most common.

1. Incenter – Located at intersection of the angle bisectors.

2. Circumcenter – Located at intersection of the perpendicular bisectors of the sides.

3. Centroid – Located at intersection of medians.

4. Orthocenter – Located at intersection of the altitudes of the triangle.

In the case of an equilateral triangle, all four of the above centers occur at the same point.

The Euler Line – It turns out that the orthocenter, centroid, and circumcenter of any triangle are collinear – that is, they always lie on the same straight line called the Euler Line, named after its discoverer – Leonhard Paul Euler.

Orthocenter

Centroid Circumcenter