Find the coordinates of the orthocenter of a triangle with the vertices (0,0), (8,2), (2,8) at each set of points on a coordinate plane?

coordinates of the orthocenter = (16/5, 16/5)

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(0,0), B(8,2), C(2,8).

Thus;

Slope of AC; m = (8 - 0)/(2 - 0)

m = 4

Since BE is perpendicular to AC, slope of BE = -1/slope of AC = -1/4

Thus, equation of BE is;

(y - 2) = -¼(x - 8)

Multiply through by 4 to get;

4y - 8 = -x + 8

x + 4y = 16

Slope of BC is; m = (8 - 2)/(2 - 8) = 6/-6 = -1

AD is perpendicular to BC, thus slope of AD = -1/-1 = 1

Equation of AD is;

(y - 0) = 1(x - 0)

y = x

Putting x for y in equation of BE, we have;

x + 4x = 16

5x = 16

x = 16/5

Since y = x in equation AD, then y = 16/5

coordinates of the orthocenter = (16/5, 16/5)

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