Question

the vertices of triangle ABC are A(-2, 2), B(4, 4), and C(1, -2). Find the coordinates of the orthocenter of triangle ABC. ​

Answer 1

The orthocenter of triangle ABC can be found by calculating the slopes of the altitudes of the triangle, forming their equations, and then finding the intersection point of any two altitudes.

Step-by-step explanation:

The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. To find the orthocenter of triangle ABC with given vertices A(-2, 2), B(4, 4), and C(1, -2), we first need to find the equations of the altitudes. Then we find the intersection point of two of these altitudes, which will be the orthocenter. Here is a step-by-step process to solve it:

1. First, find the slopes of the sides of the triangle by using the formula for slope between two points, which is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
2. Since the altitudes are perpendicular to the sides, use the negative reciprocal of the slope of each side to find the slope of each altitude.
3. Use the point-slope form, \(y - y_1 = m(x - x_1)\), to write the equation of each altitude using the slope found in step 2 and passing through the opposite vertex.
4. Find the intersection point of any two altitudes; this point is your orthocenter.

This process involves using algebra and understanding the basic properties of geometric figures on the Cartesian plane. Calculations may be extensive and are not shown in this summary.

Answer 2

(1; 4/3)

Step-by-step explanation:

for coordinate 'x':

(-2+4+1)/3=1

for coordinate 'y':

(2+4-2)/3=4/3