The height of a triangle is a fundamental geometric concept. It is defined as the perpendicular line segment drawn from any vertex of the triangle to the opposite side, or to the extension of the opposite side. In this example, we see triangle ABC where the height is drawn from vertex C to side AB, creating a right angle at point H.
Every triangle has exactly three heights, one from each vertex to the opposite side. These three heights are special because they always intersect at a single point called the orthocenter. Let me show you all three heights: the red height from vertex C, the green height from vertex A, and the purple height from vertex B. Notice how they all meet at point O, the orthocenter.
The position of triangle heights depends on the triangle type. In an acute triangle, where all angles are less than 90 degrees, all three heights fall inside the triangle. In a right triangle, two of the heights are actually the legs of the triangle themselves, since they are already perpendicular to each other. In an obtuse triangle, which has one angle greater than 90 degrees, two heights fall outside the triangle and must be drawn to the extensions of the sides.
The height of a triangle is essential for calculating its area. The area formula is one-half times base times height, where the height must be perpendicular to the chosen base. In this example, we have a triangle with a base of 4 units and a height of 2.5 units. Using the formula, the area equals one-half times 4 times 2.5, which gives us 5 square units. This formula works for any triangle, regardless of its shape.
Let's summarize what we've learned about triangle heights. First, a height is always perpendicular to the base. Second, every triangle has exactly three heights, one from each vertex. Third, these three heights always intersect at a single point called the orthocenter. Fourth, heights are essential for calculating the area using the formula: area equals one-half times base times height. Finally, the position of heights varies depending on whether the triangle is acute, right, or obtuse. Understanding triangle heights is fundamental to many geometric calculations and proofs.