Find the coordinates of the orthocenter of a triangle with the vertices (5,3), (8,6), (0,14) at each set of points on a coordinate plane?

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)

answer for #1 is c

answer for #2 is d

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a)(x-2)^2+(y-1)^2=9

step-by-step explanation:

(picture explains)