Final answer:
The orthocenter of the triangle with points (0,0), (6,3), and (8,9) is (8,4).
Step-by-step explanation:
The orthocenter of a triangle is the point at which the three altitudes of the triangle intersect. To find the orthocenter of the triangle with points (0,0), (6,3), and (8,9), we need to find the slopes of the three sides of the triangle and then find the equations of the altitudes. Finally, we can find the point of intersection of the altitudes, which will be the orthocenter.
To find the slopes of the sides, we use the formula: slope = (y2 - y1) / (x2 - x1). Using the given points, we find:
- slope of side AB: (3 - 0) / (6 - 0) = 3/6 = 1/2
- slope of side BC: (9 - 3) / (8 - 6) = 6/2 = 3
- slope of side AC: (9 - 0) / (8 - 0) = 9/8
To find the equations of the altitudes, we use the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the side and m is the slope. The equations of the altitudes are:
- Equation of altitude from A to side BC: y - 0 = 1/2(x - 0) => y = (1/2)x
- Equation of altitude from B to side AC: y - 3 = 3(x - 6) => y = 3x - 15
- Equation of altitude from C to side AB: y - 9 = (9/8)(x - 8) => y = (9/8)x - 9/2
To find the orthocenter, we find the point of intersection of the altitudes. Solving the equations, we get x = 8 and y = 4. Therefore, the orthocenter of the triangle is (8, 4).