Question

Determine whether the orthocenter is inside, on, or outside the triangle formed by the points L(-3, 2), M(5, 2), and N(-3, 6). Then find the coordinates of the orthocenter.

A) Inside; Orthocenter coordinates: (0, 5)
B) On; Orthocenter coordinates: (5, 4)
C) Outside; Orthocenter coordinates: (-2, 3)
D) Inside; Orthocenter coordinates: (5, 14)

Answer 1

The orthocenter of the triangle formed by points L(-3, 2), M(5, 2), and N(-3, 6) is inside the triangle, and the coordinates of the orthocenter are (-3, 2).

Step-by-step explanation:

The question asks us to locate the orthocenter of a triangle formed by points L(-3, 2), M(5, 2), and N(-3, 6). The orthocenter is the point where the three altitudes of the triangle meet. An altitude is a perpendicular segment from a vertex to the line containing the opposite side.

In this case, we can immediately recognize that line LM is horizontal because both points have the same y-coordinate. Similarly, LN is vertical because the x-coordinates are the same. Therefore, the altitudes dropping from vertices M and N are the very lines LN and LM respectively, since they are already perpendicular to the opposite sides. This implies that the point L, being the intersection of lines LN and LM, is the orthocenter of the triangle.

Therefore, the orthocenter is inside the triangle, and its coordinates are the same as point L, which is (-3, 2).