What is the orthocenter of triangle ABC

Question

If A(-3,-7) B(5,1) C(6,-4)

Answer 1

An orthocenter is the point where the altitudes all meet.

The altitudes always are on a vertex of the triangle, and is perpendicular to the opposite of that vertex.

Rephrasing that, the altitude is defined by a vertex(point) and the being perpendicular to the opposite side(slope). If we can find the slope and the point, we can find an equation. If we find two equations of two altitudes, the point where they meet is the orthocenter.

1. First equation
2. Randomly choose side AB and the vertex C. We need to find the slope of line AB first.
3. Slope equation: (y2-y1)/(x2-x1) → (1- -7) / (5 - -3) = 8/8 = 1
4. The altitude is perpendicular. So we want the negative reciprocal to get the slope of the perpendicular line. 1 → -1
5. The equation of a line is y = mx + b.
6. We plug in the slope first: y = -x+b
7. Then we plug in either a point on the line to find b. We know vertex C must be on the altitude by definition: -4 = -(6) + b → b = 2
8. The final equation is y = -x+2
9. Second equation
10. Randomly chose side BC and vertex A this time. We need the slope of line BC
11. slope equation: (-4 - 1)/ (6-5) = -5
12. We want the perpendicular slope though. The negative reciprocal of -5 → 1/5
13. The slope intercept equation y = mx + b
14. plug in the slope: y = (1/5)x+b
15. Plug in a point that we know is on the altitude. Since the side of the triangle we work with is BC, vertex A is on the altitude: -7 = (1/5)(-3)+b → -7 +(3/5) = -6.4=b
16. Final equation: y = (1/5)x -6.4
17. With two equations, we now need to find the intersection.
18. Set the equations equal:
19. -x+2 = (1/5)x - 6.4
20. -x-(.2)x = -6.4 - 2
21. -1.2x = -8.4
22. x = 7
23. Plug x = 7 into either of our altitude equations
24. y = -(7) + 2
25. y = -5

The orthocenter is on the point (7,-5)