Question

Find an ordered pair for the orthocenter of the triangle with vertices M(-6, -2), N(2, 6), and P(4, 0).

Answer 1

The orthocenter of the triangle with vertices M(-6, -2), N(2, 6), and P(4, 0) is located at the coordinates (2,2).

Step-by-step explanation:

The orthocenter of a triangle is the point where the altitudes of the triangle intersect. To find the orthocenter, we need to find the equations of the altitudes and solve for their point of intersection. Let's start by finding the equations of the altitudes.

The altitude from vertex M is perpendicular to side NP, which has the slope -(6-0)/(2-4) = 3. The equation of the line passing through M with a slope of -1/3 is y-(-2) = -1/3(x-(-6)). Simplifying this equation gives y = -x + 4.

The altitude from vertex N is perpendicular to side MP, which has the slope -(0-(-2))/(4-(-6)) = 2/5. The equation of the line passing through N with a slope of -5/2 is y-6 = -5/2(x-2). Simplifying this equation gives y = -5x/2 + 11.

The altitude from vertex P is perpendicular to side MN, which has the slope -(6-(-2))/(2-(-6)) = 8/4 = 2. The equation of the line passing through P with a slope of -1/2 is y-0 = -1/2(x-4). Simplifying this equation gives y = -x/2 + 2.

To find the coordinates of the orthocenter, we need to solve the system of equations formed by the equations of the altitudes. Substituting one equation into another, we find that the orthocenter has coordinates (2,2).

Answer 2

To find the orthocenter of a triangle with vertices M(-6, -2), N(2, 6), and P(4, 0), find the slopes of the altitudes passing through each vertex, find equation lines of the altitudes, and solve the system of equations to find the orthocenter at (2, 2).

Step-by-step explanation:

To find the orthocenter of a triangle with vertices M(-6, -2), N(2, 6), and P(4, 0), we need to find the point where the altitudes of the triangle intersect.

First, find the slopes of the lines passing through each pair of vertices: MN, NP, and MP. The perpendicular slopes to these lines will be the slopes of the altitudes passing through each corresponding vertex.

Next, find the equations of the lines using the vertex and slope of each altitude. Finally, solve the system of equations formed by the three altitude lines to find the point of intersection, which will be the orthocenter of the triangle.

In this case, the orthocenter is the point (2, 2).