You are given a triangle ABC with vertices A (-2, 5) and B (4, 3), and orthocenter Q (1, 2).
Determine the coordinates of vertex C.
Solution to the Problem:
Here is the solution:The coordinates of vertex C are (0, -1).
The orthocenter is the point where the three altitudes of a triangle intersect.
The altitude is the line segment drawn from one vertex of a triangle perpendicular to the opposite side. So, the product of the slopes of the altitude and the opposite side equals -1.
Let (x, y) represent the coordinates of vertex C.
The slope of AB is equal to the negative reciprocal of the slope of CQ.
2 / (-6) = -1 /((y - 2) / (x - 1))
y - 2 = 3x - 3
So, y = 3x - 1
Similarly, the slope of AQ is equal to the negative reciprocal of the slope of BC.
3 / (-3) = -1 / ((y-3) / (x-4))
y - 3 = x - 4
So, y = x - 1
Solve these two equations to obtain x = 0 and y = -1, the coordinates of C.
You can check the answer by verifying that the slope of each altitude is equal to the negative reciprocal of the slope of the corresponding side.
Here are the slopes of the three sides and the three altitudes:
AC: -3
AB: -1/3
BC: 1
--------
AQ: -1
BQ: 1/3
CQ: 3
Many thanks to Colin Bowey for providing this diagram:
Correctly solved by:
| 1. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 2. Davit Banana | Istanbul, Turkey |
| 3. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 4. Kushagra Chugh | Hisar, Haryana, India |
| 5. Kelly Stubblefield | Mobile, Alabama, USA |