Answer:
coordinates of the orthocenter = (8, 6)
Step-by-step explanation:
I have drawn a diagram showing this triangle with the vertices. I have also drawn altitude from B perpendicular to AC at point E.
I have also drawn altitude from from A perpendicular to BC at point D.
Now, we will find the slope of AC from the line slope equation; (y - y1) = m(x - x1)
m = (y - y1)/(x - x1)
Our coordinates are; A(5,3), B(8,6), C(0,14)
Thus;
Slope of AC; m = (14 - 3)/(0 - 5)
m = -11/5
Since BE is perpendicular to AC, slope of BE = -1/slope of AC = -1/(-11/5) = 5/11
Thus, equation of BE is;
(y - 6) = (5/11)(x - 8)
Multiply through by 11 to get;
11y - 66 = 5x - 40
11y - 5x = 66 - 40
11y - 5x = 26
Slope of BC is; m = (14 - 6)/(0 - 8) = 8/-6 = -1
AD is perpendicular to BC, thus slope of AD = -1/-1 = 1
Equation of AD is;
(y - 3) = 1(x - 5)
y - 3 = x - 5
y = x - 5 + 3
y = x - 2
Putting x - 2 for y in equation of BE, we have;
11(x - 2) - 5x = 26
11x - 22 - 5x = 26
6x - 22 = 26
6x = 26 + 22
6x = 48
x = 48/6
x = 8
Put 8 for x in equation AD, then y = 8 - 2 = 6
coordinates of the orthocenter = (8, 6)
