Final answer:
To find the orthocenter of a triangle, altitudes are drawn from each vertex. For the given right triangle JKL, the orthocenter should be at the right angle vertex which is (5, -2), unless there is additional context implying otherwise.
Step-by-step explanation:
To find the coordinates of the orthocenter of a triangle, you need to determine the points where the altitudes of the triangle intersect. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. In a triangle with vertices J(3, -2), K(5,6), and L(9,-2), we can see by inspection that segment JL is horizontal, and thus the altitudes from vertices K and L will be vertical lines because they must be perpendicular to JL. Since L is already on JL, the altitude from L is simply JL itself.
The altitude from K will be a vertical line because JL is horizontal. Knowing the x-coordinate of K is 5, the equation of this altitude will be x = 5. To find where this altitude intersects JL, we look for the intersection point with the same x-coordinate as K, which would be (5, y). Since J and L have the same y-coordinate (-2), this confirms that the altitude from K intersects JL at the y-coordinate -2.
Therefore, these two altitudes intersect at the point (5, -2). However, the statement from the geometry teacher gives the orthocenter as (5, -1). This might be an error unless there's additional context or a specific property of the triangle that wasn't provided. Typically, the orthocenter can be outside the triangle for obtuse triangles, but in this case, the given points form a right triangle, and the orthocenter should lie at the vertex of the right angle. Assuming no further context, the orthocenter appears to be at (5, -2).