Answer:
y = 2x - 4
Step-by-step explanation:
To solve this problem, we must first calculate the slope of the line AB using the formula:
[tex]\boxed{m = \frac{y_2 - y_1}{x_2 - x_1}}[/tex]
where:
m ⇒ slope of the line
(x₁, y₁), (x₂, y₂) ⇒ coordinates of two points on the line
Therefore, for line AB with points A = (2, 5) and B = (4, 4) :
[tex]m_{AB} = \frac{5 - 4}{2 - 4}[/tex]
⇒ [tex]m_{AB} = \frac{1}{-2}[/tex]
⇒ [tex]m_{AB} = -\frac{1}{2}[/tex]
Next, we have to calculate the slope of the line BC.
We know that the product of the slopes of two perpendicular lines is -1.
Therefore:
[tex]m_{BC} \times m_{AB} = -1[/tex] [Since BC and AB are at right angles to each other]
⇒ [tex]m_{BC} \times -\frac{1}{2} = -1[/tex]
⇒ [tex]m_{BC} = -1 \div -\frac{1}{2}[/tex] [Dividing both sides of the equation by -1/2]
⇒ [tex]m_{BC} = \bf 2[/tex]
Next, we have to use the following formula to find the equation of line BC:
[tex]\boxed{y - y_1 = m(x - x_1)}[/tex]
where (x₁, y₁) are the coordinates of a point on the line.
Point B = (4, 4) is on line BC, and its slope is 2. Therefore:
[tex]y - 4 =2 (x - 4)[/tex]
⇒ [tex]y - 4 = 2x - 8[/tex] [Distributing 2 into the brackets]
⇒ [tex]y = 2x-4[/tex]
Therefore, the equation of line BC is y = 2x - 4.
