Answer:
y = ¹⁴/₄.x
Step-by-step explanation:
First of, if the bisector is perpendicular to the line segment, then we can find the gradient of the bisector ([tex]m_{b}[/tex]) using the rule/principle:
Let:
m = gradient of the line segment
Then:
[tex]m_{b}[/tex] = [tex]-\frac{1}{m}[/tex]
We can find m since we have two points that fall on the line segment, (5, -9) and (-9, -5):
[tex]m =[/tex] Δy/Δx
[tex]m = \frac{-9 - (-5)}{5 - (-9)} \\\\ m = -\frac{4}{14}[/tex]
We can now find [tex]m_{b}[/tex]:
[tex]m_{b} = -\frac{1}{(-\frac{4}{14}) } \\\\ m_{b} = 1.\frac{14}{4} \\\\ m_{b} = \frac{14}{4}[/tex]
The equation of a line can be found using:
y - y₁ = m(x - x₁)
We have the gradient of the perpendicular bisector, the only other thing we need to identify the equation of the bisector is coordinates of a point that fall on the line;
We know the line will pass through the point exactly midway between (5, -9) and (-9, -5) since it is a bisector;
This can be found by:
[tex]x_{1} = -9 + \frac{5 - (-9)}{2} \\ x_{1} = -9 + 7 \\ x_{1} = -2 \\\\ y_{1} = -5 + \frac{-9 - (-5)}{2} \\ y_{1} = -5 +(-2) \\ y_{1} = -7 \\\\ (-2, -7)[/tex]
We have a point on the line and the gradient so we can now find the equation:
[tex]y - (-7) = \frac{14}{4}(x - (-2)) \\\\ y + 7 = \frac{14}{4}(x + 2) \\\\ 4y + 28 = 14(x + 2) \\\\ 4y + 28 = 14x + 28 \\\\ 4y = 14x \\\\ y = \frac{14x}{4}[/tex]
