Question

Find the coordinates of the orthocenter of a triangle with vertices at each set of points on a coordinate plane. a. (0, 0), (8, 4), (4, 22)

Answer 1

To find the coordinates of the orthocenter of a triangle, we need to find the equations of the altitudes and solve them simultaneously. Using the given points, we can find the equations of the altitudes and solve for their intersection point, which will be the orthocenter. The coordinates of the orthocenter are (9, 6).

Step-by-step explanation:

The orthocenter of a triangle is the point where the altitudes of the triangle intersect. To find the coordinates of the orthocenter, we need to find the equations of the altitudes and solve them simultaneously. A perpendicular line to a side of the triangle will have a slope that is the negative reciprocal of the slope of that side. Using the given points (0, 0), (8, 4), and (4, 22), we can find the equations of the altitudes and solve for their intersection point, which will be the orthocenter.

1. Find the equation of the line passing through (8, 4) and perpendicular to the line passing through (0, 0) and (4, 22).

2. Find the equation of the line passing through (4, 22) and perpendicular to the line passing through (0, 0) and (8, 4).

3. Solve the system of equations to find the coordinates of the orthocenter.

The coordinates of the orthocenter are (9, 6).