The orthocenter describes the point of concurrency for the lines containing the altitudes of a triangle.
Inside Orthocenter: (3,5.2)
Practice makes perfect
Let's begin by drawing the triangle using the given coordinates.
To find the location of the orthocenter, we need to recall two definitions.
The orthocenter describes the point of concurrency for the lines containing the altitudes of a triangle.
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 GH and GJ.
Equation of the Altitude of GH
Since GH is horizontal, the altitude will be vertical. From the diagram, we can see that it is a vertical line passing through J( 3,1) Therefore, the equation of the line for the line segment of the altitude is x= 3.
Equation of the Altitude of GJ
To find the equation of the second altitude, we need the slope of GJ. We can use the Slope Formula and the coordinates of G and J to do this.
We found that the slope of GJ is - 52. 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.
The slope of the altitude is 25.
y=2/5x+b
From the diagram, we also know that the altitude passes through the vertex H(5,6). Let's substitute these coordinates to find the y-intercept.
