Find the coordinates of the orthocenter of a triangle with vertices at each set of points on a coordinate plane. a. (0,0), (16,4), (4,6) b. (3,4), (11,12), (8,15)

The answer is below

Step-by-step explanation:

a) The equation of a line passing through the point (0,0) and (16,4) is given by:

We have to find the equation of the line perpendicular to y = (1/4)x and passing through the point (4,6).

The line perpendicular to y = (1/4)x has a slope of -4 (product of their slope = -1).

Hence:

The slope of a line passing through the point (16,4) and (4,6) is given by:

We have to find the equation of the line perpendicular to  the line with slope of -1/6 and passing through the point (0,0).

The line perpendicular to a line with slope -1/6 has a slope of 6

Hence:

Solving y = 6x and y  = -4x + 22, gives:

x = 2.2, y = 13.2

Hence the orthocenter is at (2.2, 13.2)

b)

The slope of a line passing through the point (3,4) and (11,12) is given by:

We have to find the equation of the line perpendicular to  the line with slope of 1 and passing through the point (8,15).

The line perpendicular to a line with slope 1 has a slope of -1

Hence:

The slope of a line passing through the point (11,12) and (8,15) is given by:

We have to find the equation of the line perpendicular to  the line with slope of -1 and passing through the point (3,4).

The line perpendicular to a line with slope -1 has a slope of 1

Hence:

Solving y = -x + 23 and y  = x + 1, gives:

x = 11, y = 12

Hence the orthocenter is at (11, 12)

4.

step-by-step explanation:

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