Answer: Circumcenter =
Orthocenter = (-4, -6)
<u>Step-by-step explanation for Circumcenter:</u>
Step 1: Find the midpoint of a line:<em> I chose (-4, 3) and (-4, -6)</em>
![\bigg(\dfrac{-4-4}{2},\dfrac{3-6}{2}\bigg)=\bigg(\dfrac{-8}{2},\dfrac{-3}{2}\bigg) = \bigg(-4,-\dfrac{3}{2}\bigg)](https://tex.z-dn.net/?f=%5Cbigg%28%5Cdfrac%7B-4-4%7D%7B2%7D%2C%5Cdfrac%7B3-6%7D%7B2%7D%5Cbigg%29%3D%5Cbigg%28%5Cdfrac%7B-8%7D%7B2%7D%2C%5Cdfrac%7B-3%7D%7B2%7D%5Cbigg%29%20%3D%20%5Cbigg%28-4%2C-%5Cdfrac%7B3%7D%7B2%7D%5Cbigg%29)
Step 2: Find the perpendicular line that passes through that point:
Since it is a vertical line, the perpendicular line is ![y=-\dfrac{3}{2}](https://tex.z-dn.net/?f=y%3D-%5Cdfrac%7B3%7D%7B2%7D)
Step 3: repeat Steps 1 and 2 for another line: <em> chose (-4, -6) and (6, -6)</em>
![\bigg(\dfrac{-4+6}{2},\dfrac{-6-6}{2}\bigg)=\bigg(\dfrac{2}{2},\dfrac{-12}{2}\bigg) = (1,-6)](https://tex.z-dn.net/?f=%5Cbigg%28%5Cdfrac%7B-4%2B6%7D%7B2%7D%2C%5Cdfrac%7B-6-6%7D%7B2%7D%5Cbigg%29%3D%5Cbigg%28%5Cdfrac%7B2%7D%7B2%7D%2C%5Cdfrac%7B-12%7D%7B2%7D%5Cbigg%29%20%3D%20%281%2C-6%29)
Since it is a horizontal line, the perpendicular line is: x = 1
Step 4: Find the intersection of the two lines ![\bigg(y=-\dfrac{3}{2}\ \text{and}\ x = 1\bigg)](https://tex.z-dn.net/?f=%5Cbigg%28y%3D-%5Cdfrac%7B3%7D%7B2%7D%5C%20%5Ctext%7Band%7D%5C%20x%20%3D%201%5Cbigg%29)
Their point of intersection is: ![\bigg(1, -\dfrac{3}{2}\bigg)](https://tex.z-dn.net/?f=%5Cbigg%281%2C%20-%5Cdfrac%7B3%7D%7B2%7D%5Cbigg%29)
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<u>Step-by-step explanation for Orthocenter:</u>
Step 1: Find the perpendicular slope of a line: <em>I chose (-4, 3) and (-4, -6)</em>
Slope is undefined. Perpendicular slope is 0.
Step 2: Use the Point-Slope formula to find the equation of the line that passes through the vertex that is opposite of the line from Step 1 and has the perpendicular slope (found in Step 1).
Vertex (6, -6) and m⊥ = 0 ⇒ y + 6 = 0(x - 6) ⇒ y = -6
Step 3: repeat Steps 1 and 2 for another line: <em> chose (-4, -6) and (6, -6)</em>
Slope is 0. Perpendicular slope is undefined (x = __ )
Vertex (-4, 3) and m⊥ = undefined ⇒ x = -4
Step 4: Find the intersection of the two lines ![(y=-6\ \text{and}\ x = -4)](https://tex.z-dn.net/?f=%28y%3D-6%5C%20%5Ctext%7Band%7D%5C%20x%20%3D%20-4%29)
Their point of intersection is: (-4, -6)