Answer: Circumcenter = [tex]\bold{\bigg(1,-\dfrac{3}{2}\bigg)}[/tex] Orthocenter = (-4, -6)
Step-by-step explanation for Circumcenter:
Step 1: Find the midpoint of a line: I chose (-4, 3) and (-4, -6)
[tex]\bigg(\dfrac{-4-4}{2},\dfrac{3-6}{2}\bigg)=\bigg(\dfrac{-8}{2},\dfrac{-3}{2}\bigg) = \bigg(-4,-\dfrac{3}{2}\bigg)[/tex]
Step 2: Find the perpendicular line that passes through that point:
Since it is a vertical line, the perpendicular line is [tex]y=-\dfrac{3}{2}[/tex]
Step 3: repeat Steps 1 and 2 for another line: chose (-4, -6) and (6, -6)
[tex]\bigg(\dfrac{-4+6}{2},\dfrac{-6-6}{2}\bigg)=\bigg(\dfrac{2}{2},\dfrac{-12}{2}\bigg) = (1,-6)[/tex]
Since it is a horizontal line, the perpendicular line is: x = 1
Step 4: Find the intersection of the two lines [tex]\bigg(y=-\dfrac{3}{2}\ \text{and}\ x = 1\bigg)[/tex]
Their point of intersection is: [tex]\bigg(1, -\dfrac{3}{2}\bigg)[/tex]
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Step-by-step explanation for Orthocenter:
Step 1: Find the perpendicular slope of a line: I chose (-4, 3) and (-4, -6)
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: chose (-4, -6) and (6, -6)
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 [tex](y=-6\ \text{and}\ x = -4)[/tex]
Their point of intersection is: (-4, -6)
