Question

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​)

Answer 1

Answer:

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:

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)\\\\y-0=\frac{4-0}{16-4}(x-0)\\\\y=\frac{1}{4}x$

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:

$y- y_1=m(x-x_1)\\\\y-6=-4(x-4)\\\\y-6=-4x+16\\\\y=-4x+22$

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

$m=\frac{y_2-y_1}{x_2-x_1}\\\\m=\frac{6-4}{4-16}=-\frac{1}{6}$

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:

$y- y_1=m(x-x_1)\\\\y-0=6(x-0)\\\\y=6x$

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:

$m=\frac{y_2-y_1}{x_2-x_1}\\\\m=\frac{12-4}{11-3}=1$

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:

$y- y_1=m(x-x_1)\\\\y-15=-1(x-8)\\\\y=-x+8+15\\\\y=-x+23$

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

$m=\frac{y_2-y_1}{x_2-x_1}\\\\m=\frac{15-12}{8-11}=-1$

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:

$y- y_1=m(x-x_1)\\\\y-4=1(x-3)\\\\y=x-3+4\\\\y=x+1$

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

x = 11, y = 12

Hence the orthocenter is at (11, 12)