Question 1103165: find the orthocenter of a triangle with the given vertices:
U(-4,-9), V(-4,6),W(5,-3)
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! In our example, we will use the following coordinates as the vertices of the triangle. A (3, 1) B(2, 2) C (3, 5)
Find the equations of 2 segments of the triangle (for our example we will find the equations for AB, and BC)
Once you have the equations from step #1, you can find the slope of the corresponding perpendicular lines.
You will use the slopes you have found from step #2, and the corresponding opposite vertex to find the equations of the 2 lines.
Once you have the equation of the 2 lines from step #3, you can solve the corresponding x and y, which is the coordinates of the orthocenter.
The steps might seem daunting, but once you actually work through the problem, you will see that it is a very easy process.
Step 1: Find equations of the line segments AB and BC.
To find any line segment, you will need to find the slope of the line and then the corresponding y-intercept.
A (3, 1) B(2, 2) C (3, 5)
Slope of AB = (1-2)/(3-2) = -1/1 = -1
y = mx + b (substitute m = -1, x = 3, y = 1)
1 = -1(3) + b
b = 4
Equation of AB: y = -1x + 4
Slope of BC = (2-5)/(2-3) = -3/-1 = 3
y = mx + b (substitute m = 3, x = 2, y = 2)
2 = 3(2) + b
b = -4
Equation of BC: y = 3x - 4
Step 2: Find the slope of the corresponding perpendicular lines
Slope of AB = -1
Slope of perpendicular line to AB: -1*m = -1 -> m = 1
Slope of BC = 3
Slope of perpendicular line to BC: 3*m = -1 -> m = -1/3
Step 3: Find the equation of the perpendicular lines
Slope of perpendicular line to AB: m = 1
We will use the coordinate of the opposite vertex (point C) to find the equation of the line.
y = mx + b (substitute m = 1, x = 3, y = 5)
5 = 1(3) + b
b = 2
Equation of perpendicular line to AB: y = 1x + 2
Slope of perpendicular line to BC: m = -1/3
We will use the coordinate of the opposite vertex (point A) to find the equation of the line.
y = mx + b (substitute m = -1/3, x = 3, y = 1)
1 = -1/3*(3) + b
b = 2
Equation of perpendicular line to AB: y = -1/3x + 2
Step 4: solve 2 perpendicular lines
equation 1: y = 1x + 2
equation 2: y = -1/3x + 2
Solving for x and y:
1x + 2 = -1/3x + 2
4/3x = 0
x = 0
y = 1(0) + 2
y = 2
The coordinates are (0, 2). This is the orthocenter.
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