Exercise 15 Page 464

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

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.

We can see that the altitudes intersect inside the triangle. Therefore, the orthocenter lies inside the triangle. To find its coordinates, we should determine the equations for two of the altitudes and solve the system of these equations. Let's use the altitudes of DE and EF.

Equation of the Altitude of DE

Since DE is vertical, its altitude will be horizontal. From the diagram, we can see that OF is a horizontal line through y= 3. Therefore, the equation of the line for the line segment of the altitude is y=3.

Equation of the Altitude of EF

To find the equation for the second altitude, we need the slope of EF. We can use the Slope Formula and the coordinates of E and F to do this. We found that the slope of EF is - 23. The product of the slopes of two perpendicular lines is -1. This allows us to find the slope of the altitude. Let's call it m_a. - 23 * m_a = -1 ⇒ m_a = 3/2 The slope of the altitude is 32. From the diagram, we also know that the altitude passes through the point D(0,0). This is a y-intercept. Therefore, the equation becomes y= 32x.

Solving for the Coordinates

Finally, we can solve the system of the found equations to find the coordinates of their intersection. Therefore, the coordinates of the orthocenter are (2,3).