Exercise 12 Page 328

Let's begin by drawing the triangle using the given coordinates.

Notice that ZY is vertical and XY is horizontal. This means that ∠ Y is a right angle. Therefore, △ XYZ is a right triangle.

To find the location of the orthocenter, we need to recall two definitions.

1. The orthocenter describes the point of concurrency for the lines containing the altitudes of a triangle.
2. An altitude of a triangle is the perpendicular segment from a vertex to the opposite side of a triangle or to the line containing the opposite side.

Let's draw the altitudes of the vertices of our triangle.

Notice that the legs XY and ZY are also altitudes. It seems that along with the third altitude they intersect at the vertex Y. Since the legs intersect at vertex, we know that the point of concurrency of the altitudes is exactly at the vertex Y. Therefore, the orthocenter is on the triangle and its coordinates are (7,-4).