Final answer:
The orthocenter of the triangle formed by the points (4, -3), (8, -5), and (8, -8) is (-1, -76/5).
Step-by-step explanation:
The orthocenter of a triangle is the point where all three altitudes of the triangle intersect.
To find the orthocenter, we need to find the altitudes of the triangle formed by the given points.
An altitude is a perpendicular segment from a vertex of the triangle to the opposite side.
We can find the equation of the line passing through two points and then find the equation of the line perpendicular to it passing through the third point.
The intersection of these two lines will give us the orthocenter of the triangle.
Let's find the equations of the lines:
- Line passing through (4, -3) and (8, -5):
m = (-5 - (-3)) / (8 - 4) = -2 / 4 = -1/2
Using the point-slope form of a line, y - y1 = m(x - x1), we get:
y - (-3) = -1/2(x - 4)
y + 3 = -1/2x + 2
y = -1/2x - 1 - Line passing through (4, -3) and (8, -8):
m = (-8 - (-3)) / (8 - 4) = -5 / 4
Using the point-slope form of a line, y - y1 = m(x - x1), we get:
y - (-3) = -5/4(x - 4)
y + 3 = -5/4x + 5
y = -5/4x + 2
Now, let's find the equation of the line perpendicular to the second line passing through the point (8, -8).
The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line.
Therefore, the slope of the perpendicular line is 4/5. Using the point-slope form of a line, we get:
y - (-8) = 4/5(x - 8)
y + 8 = 4/5x - 32/5
y = 4/5x - 72/5
Finally, let's find the intersection point of the two lines to determine the orthocenter:
-1/2x - 1 = 4/5x - 72/5
-5/10x - 1 + 72/5 = 8/10x
-5/10x + 72/5 - 50/10 = 8/10x
-5/10x - 8/10x = -22/10x = -11/5x
-22/10x = -22/10
x = -1
Substituting the value of x into one of the equations, we get:
y = 4/5(-1) - 72/5
y = -4/5 - 72/5
y = -76/5
Therefore, the orthocenter of the triangle formed by the points (4, -3), (8, -5), and (8, -8) is (-1, -76/5).