Final answer:
To determine the orthocenter of the triangle formed by the points L(-3, 2), M(5, 2), and N(-3, 6), find the equations of the altitudes from each point and solve the system of equations to find their intersection points to determine the relative position of the orthocenter to the triangle.
Step-by-step explanation:
To determine whether the orthocenter is inside, on, or outside the triangle, we need to find the altitudes of the triangle and see where they intersect.
Let's start with the altitude from point L.
The altitude passes through the opposite side, which is the line segment MN.
The slope of MN is (6-2)/(-3-5) = 4/-8 = -1/2.
The equation of the line passing through L with a slope of -1/2 is y = -1/2x + 1.
To find the intersection point with MN, we solve the system of equations:
y = -1/2x + 1 (equation of the altitude)
y = 2 (equation of MN)
Solving these equations, we get x = 4 and y = 2. So, the altitude from L intersects MN at the point (4, 2).
Similarly, you can find the equations of the altitudes from points M and N and solve the system of equations to find their intersection points.
Then, determine the relative position of the orthocenter to the triangle.