Medians, Centroids, Altitudes, and Orthocenters

Front

What is a median in a triangle?

Back

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

Front

What is a centroid?

Back

The centroid is the point where all three medians of a triangle intersect. It is also the center of mass of the triangle.

Front

What is an altitude in a triangle?

Back

An altitude is a perpendicular segment from a vertex to the line containing the opposite side.

Front

What is an orthocenter?

Back

The orthocenter is the point where all three altitudes of a triangle intersect.

Front

What is the relationship between the centroid and the segments it creates?

Back

The centroid divides each median into two segments, with the segment connecting the vertex to the centroid being twice as long as the segment connecting the centroid to the midpoint.

Front

How do you find the circumcenter of a triangle?

Back

The circumcenter can be found by constructing the perpendicular bisectors of at least two sides of the triangle; their intersection is the circumcenter.

Front

What is the significance of the circumcenter?

Back

The circumcenter is equidistant from all three vertices of the triangle and is the center of the circumcircle.

Front

If a triangle has sides of lengths 3, 4, and 5, what type of triangle is it?

Back

It is a right triangle, as it satisfies the Pythagorean theorem (3^2 + 4^2 = 5^2).

Front

What is the formula to find the area of a triangle using the base and height?

Back

Area = 1/2 * base * height.

Front

What is the relationship between the orthocenter and the type of triangle?

Back

In an acute triangle, the orthocenter is inside the triangle; in a right triangle, it is at the vertex of the right angle; and in an obtuse triangle, it is outside the triangle.