why is the orthocenter of a right triangle on the vertex that is a right angle?
The orthocenter of any triangle is the point where the three altitudes of the triangle intersect.
In the case of a right triangle, it is on the altitude from the right angle vertex to the hypotenuse.
Altitudes are perpendicular to one of the sides of a triangle and pass through the vertex opposite to that line. Perpendicular lines form right angles. A right triangle has a right angle. That means that the two line segments making up that angle are perpendicular to each other, so they are altitudes. The remaining altitude (which is perpendicular to the hypotenuse) passes through the vertex of the right angle, since they are opposite of each other. All three altitudes pass through the vertex of the right angle.
(Hopefully the wording wasn't too confusing.)
The orthocenter of a triangle is the point where all three altitudes of the triangle intersect. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side.
In the case of a right triangle, one of the angles is a right angle, which means one of the sides is horizontal or vertical. Let's consider a right triangle where the right angle is at vertex C, and vertices A and B are the other two vertices.
To find the orthocenter, we need to draw the altitudes from each of the vertices. The altitude from vertex C will be a vertical line segment passing through C and intersecting the opposite side (AB) at a point, say D.
Since the right angle is at vertex C, the opposite side (AB) is the hypotenuse of the right triangle. The altitude from vertex C, therefore, falls perpendicularly onto the hypotenuse, dividing it into two segments, CD and BD.
Now, let's analyze the altitudes from vertices A and B. Since the right triangle is not isosceles, these altitudes are not vertical or horizontal lines. Instead, they are inclined lines that form acute angles with the opposite sides (BC and AC, respectively).
When we extend these altitudes and check where they intersect with each other, we find that they intersect at point E, which is coincident with vertex C. Therefore, we have three altitudes (CD, AE, and BE) all intersecting at vertex C, which is the orthocenter.
In conclusion, for a right triangle, the orthocenter coincides with the vertex that forms the right angle because the altitude from that vertex (C) is perpendicular to the opposite side (AB), while the altitudes from the other two vertices (A and B) intersect at a point coinciding with C.