Finding the Orthocenter

This page focuses on the practical application of finding the orthocenter of a triangle using coordinate geometry.

Example: For a triangle ABC with vertices A(1,3), B(2,7), and C(6,3), we can find the orthocenter using the following steps:

1. Calculate the slopes of the sides of the triangle.
2. Determine the equations of the altitudes using the perpendicular slope property.
3. Find the intersection point of two altitudes, which will be the orthocenter.

In this case, the calculations lead to the orthocenter being located at the point (2,4).

Highlight: The process of finding the orthocenter involves applying concepts of coordinate geometry, slope calculations, and solving systems of linear equations.

Understanding how to find the orthocenter of a triangle given 3 points is a valuable skill in geometry and can be applied to various problems involving triangles and their properties.

Medians and Altitudes of Triangles

This page introduces the concepts of medians and altitudes in triangles, along with their properties and the theorems related to their concurrency.

Definition: A median is a segment that connects a vertex of a triangle to the midpoint of the opposite side.

Highlight: The point where the medians of a triangle intersect is called the centroid, also known as the center of gravity.

The Concurrency of Medians Theorem states that 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: In a triangle where XA = 8, the length of XB can be calculated as XB = XA + AB = 8 + 4 = 12.

Definition: An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.

Altitudes can be inside, outside, or coincide with a side of the triangle, depending on the triangle's shape.

The Concurrency of Altitudes Theorem states that the lines containing the altitudes of a triangle are concurrent, meaning they intersect at a single point.

Vocabulary: The point where the altitudes intersect is called the orthocenter of the triangle.