Which term describes the point where the three altitudes of a triangle intersect? A. Incenter B. Orthocenter C. Centroid D. Circumcenter
The Correct Answer and Explanation is:
Correct Answer: B. Orthocenter
The point where the three altitudes of a triangle intersect is called the orthocenter.
Explanation:
In geometry, a triangle has several important points of concurrency—points where specific types of lines intersect. These points help in understanding the triangle’s properties better.
- Altitude: An altitude of a triangle is a perpendicular line drawn from a vertex to the opposite side (or the extension of that side). Every triangle has three altitudes, one from each vertex.
- Orthocenter: The orthocenter is the point of intersection of the three altitudes of a triangle. Depending on the type of triangle, the orthocenter may lie:
- Inside the triangle for acute triangles,
- On the triangle for right triangles (specifically at the right-angled vertex),
- Outside the triangle for obtuse triangles.
- Other Points of Concurrency (Incorrect Options):
- A. Incenter: This is the point where the three angle bisectors intersect. It is the center of the inscribed circle (incircle) that touches all three sides.
- C. Centroid: This is where the three medians of a triangle intersect. A median connects a vertex to the midpoint of the opposite side. The centroid is also the triangle’s center of mass or balance point.
- D. Circumcenter: This is the point where the perpendicular bisectors of the sides intersect. It is the center of the circumscribed circle (circumcircle) that passes through all three vertices.
Summary:
Among the listed choices, the orthocenter specifically refers to the intersection of altitudes. Understanding these centers is key to solving advanced geometry problems and recognizing their roles in triangle constructions.
