Answer: [tex]y=\dfrac{9}{2}x+\dfrac{51}{4}[/tex] → y = 4.5x + 12.75
Step-by-step explanation:
Perpendicular Bisector means the line goes through the midpoint and has the opposite reciprocal slope.
Use the Midpoint formula: [tex]M=\bigg(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\bigg)[/tex]
[tex]M=\bigg(\dfrac{-6+3}{2},\dfrac{7+5}{2}\bigg)\\\\\\.\ =\bigg(\dfrac{-3}{2},\dfrac{12}{2}\bigg)\\\\\\.\ =\bigg(-\dfrac{3}{2},6\bigg)[/tex]
Use the Slope formula: [tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\dfrac{7-5}{-6-3}\quad =\quad \dfrac{2}{-9}\quad =\quad -\dfrac{2}{9}[/tex]
Perpendicular slope is opposite (change the sign) and reciprocal (flip the fraction). [tex]m_{\perp}=+\dfrac{9}{2}[/tex]
Now use the Point-Slope formula: y - y₁ = m⊥ (x - x₁) where
- (x₁, y₁) is the midpoint (-3/2, 6)
- m⊥ = 9/2
[tex]y-6=\dfrac{9}{2}\bigg(x+\dfrac{3}{2}\bigg)\\\\\\y-6=\dfrac{9}{2}x+\dfrac{27}{4}\\\\\\y\quad =\dfrac{9}{2}x+\dfrac{27}{4}+\dfrac{24}{4}\\\\\\.\quad \large\boxed{y =\dfrac{9}{2}x+\dfrac{51}{4}}[/tex]
