find the equation of the line that passes through (-2,1) and is perpendicular to AB
the points A(-2,3) and B(4,-1)
To find the equation of the line that passes through (-2,1) and is perpendicular to AB, we first need to find the slope of line AB.
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
In this case, the points A(-2,3) and B(4,-1) give us:
m = (-1 - 3) / (4 - (-2))
= (-4) / (4 + 2)
= -4 / 6
= -2/3
The slope of line AB is -2/3.
To find the slope of the line perpendicular to AB, we take the negative reciprocal of -2/3:
m_perpendicular = -1 / (-2/3)
= 3/2
Now, we can use the point-slope form of a linear equation to find the equation of the line passing through (-2,1) with slope 3/2:
y - y₁ = m(x - x₁) (where (x₁, y₁) is the given point (-2,1) and m is the slope 3/2)
Replacing the values, we have:
y - 1 = (3/2)(x - (-2))
y - 1 = (3/2)(x + 2)
Expanding and simplifying:
y - 1 = (3/2)x + 3
y = (3/2)x + 3 + 1
y = (3/2)x + 4
So, the equation of the line that passes through (-2,1) and is perpendicular to AB is y = (3/2)x + 4.