Perpendiculars from the vertices to the opposite sides of a triangle

When perpendiculars are drawn from the vertices of a triangle to the opposite sides, they meet at a point called the orthocenter. This point is significant in triangle geometry and has several important properties.

Orthocenter

The orthocenter of a triangle is the point of intersection of the three perpendiculars drawn from the vertices to the opposite sides. It is denoted by H. The orthocenter may lie inside, outside, or on the triangle, depending on the type of triangle.

Properties of the Orthocenter

1. Right Triangle: In a right triangle, the orthocenter coincides with one of the vertices of the right angle.

2. Acute Triangle: In an acute triangle, the orthocenter lies inside the triangle.

3. Obtuse Triangle: In an obtuse triangle, the orthocenter lies outside the triangle.

4. Equilateral Triangle: In an equilateral triangle, all three perpendiculars are also the altitudes, and they meet at the same point, which is the orthocenter.

Centroid, Circumcenter, and Incenter

While the orthocenter is the point of intersection of perpendiculars from the vertices to the opposite sides, there are three other important points in a triangle:

1. Centroid: The centroid is the point of intersection of the medians of a triangle. The medians are the line segments drawn from each vertex to the midpoint of the opposite side. The centroid divides each median into two segments, with the ratio of 2:1.

2. Circumcenter: The circumcenter is the center of the circumcircle, which is a circle passing through all three vertices of the triangle. It is the point equidistant from the three vertices. The perpendicular bisectors of the sides of the triangle meet at the circumcenter.

3. Incenter: The incenter is the center of the incircle, which is a circle inscribed inside the triangle. It is the point equidistant from all three sides of the triangle. The angle bisectors of the triangle meet at the incenter.

Conclusion

In the given question, the perpendiculars drawn from the vertices to the opposite sides of a triangle meet at the orthocenter. Therefore, the correct answer is option 'A' - orthocenter.