Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Where do the altitudes of a triangle meet?

1. At the circumcenter

2. At the centroid

3. At the orthocenter

4. At the incenter

The correct answer is: At the orthocenter

The altitudes of a triangle intersect at a point known as the orthocenter. In geometry, an altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. Each triangle has three altitudes, one from each vertex, which can either lie inside the triangle, or extend outside for obtuse triangles. The orthocenter's location varies depending on the type of triangle. In acute triangles, the orthocenter is located within the triangle, while in right triangles, it is found at the vertex of the right angle. In obtuse triangles, the orthocenter lies outside the triangle. This unique property distinguishes the orthocenter as the point of concurrency for the altitudes, showcasing the relationship between the triangle's angles and sides. The other points mentioned, such as the circumcenter, centroid, and incenter, represent different properties of the triangle: the circumcenter is the intersection of the perpendicular bisectors of the sides, the centroid is where the three medians intersect, and the incenter is the point where the angle bisectors meet. Each plays a significant role in triangle geometry but does not coincide with the intersection of the altitudes.