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Median and Altitude of a TriangleSec 5.3

- Goal
- To use properties of the medians of a triangle.
- To use properties of the altitudes of a triangle.

Median of a Triangle

- Median of a Triangle a segment whose endpoints

are the vertex of a triangle and the midpoint of

the opposite side.

Vertex

Median

Median of an Obtuse Triangle

Point of concurrency P or centroid

Medians of a TriangleTheorem 5.7

The medians of a triangle intersect at a point

that is two-thirds of the distance from each

vertex to the midpoint of the opposite side.

Example - Medians of a Triangle

Median of an Acute Triangle

Point of concurrency P or centroid

Median of a Right Triangle

Point of concurrency P or centroid

The three medians of an obtuse, acute, and a

right triangle always meet inside the triangle.

Altitude of a Triangle

Altitude of a triangle the perpendicular

segment from the vertex to the opposite side or

to the line that contains the opposite side

altitude

Altitude of an Acute Triangle

Point of concurrency P or orthocenter

The point of concurrency called the orthocenter

lies inside the triangle.

Altitude of a Right Triangle

The two legs are the altitudes

The point of concurrency called the orthocenter

lies on the triangle.

Point of concurrency P or orthocenter

Altitude of an Obtuse Triangle

altitude

altitude

The point of concurrency of the three altitudes

is called the orthocenter

The point of concurrency lies outside the

triangle.

Altitudes of a TriangleTheorem 5.8

The lines containing the altitudes of a triangle

are concurrent.

altitude

altitude

altitude

Example

- Determine if EG is a perpendicular bisector, and

angle bisector, a median, or an altitude of

triangle DEF given that

Review Properties / Points of Concurrency

- Median -- Centroid
- Altitude -- Orthocenter
- Perpendicular Bisector -- Circumcenter
- Angle Bisector -- Incenter

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