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The orthocenter describes the point of concurrency for the lines containing the altitudes of a triangle.
Location: Inside
Orthocenter: (0,4)
Let's begin by drawing the triangle using the given coordinates.
To find the location of the orthocenter, we need to recall two definitions.
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 TV and UV.
Since TV is horizontal, its altitude will be vertical. From the diagram, we can see that PU is a vertical line through x=0. Therefore, the equation of the line for the line segment of the altitude is x=0.
To find the equation for the second altitude, we need the slope of UV. We can use the Slope Formula and the coordinates of U and V to do this.
We found that the slope of UV is 2. 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. 2* m_a = -1 ⇒ m_a = -1/2 The slope of the altitude is - 12. y=-1/2x+b To complete its equation, we need the y-intercept. We can use the fact that the altitude passes through the vertex T(-2,5).
x= -2, y= 5
- a(- b)=a* b
a/a=1
LHS-1=RHS-1
Rearrange equation
Therefore, the equation of the altitude of UV is y=- 12x+4.
Finally, we can solve the system of the found equations to find the coordinates of their intersection.
(II): x= 0
(II): Zero Property of Multiplication
(II): Add terms
Therefore, the coordinates of the orthocenter are (0,4).
