3. Find the equation of the line that goes through the point (2, 3) perpendicular to the line whose equation is x + y = 3.
In slope intercept form, the equation of the line will be
where m is the slope and b is the y-intercept.
In problem 2, we sqaw that the slope of the line whose equation is x + y = 3 was -1. The slope of a perpendicular line would be the negative reciprocal of that or
As with problem 2, the point on the line is
We put this into the slope intercept form of the equation
We subtract 2 from both sides to solve for b, and get
so the equation of the perpendicular line is
The foot of the point is where this perpendicular line through the point meets the given line. We set up a system of simultaneous linear equations.
This system is set up best for the substitution method. Substitute the formula which is equal to y from the second equation in for the y in the first equation.
combine like terms,
subtract 1 from both sides,
and divide both sides by 2.
Now that we know that the x coordinate of the point is 1, we can substitute it in for the x in either one of the equations and get that
So the coordinates for the foot are
The graph looks like
To find the reflection of (2, 3) about the line whose equation is x + y = 3, note that the x-coordinate of the given point, 2, is 1 more than the x-coordinate of the foot, 1. As a result, the x- coordinate of the reflection will be 1 less than the x-coordinate of the foot, or 0. Since the reflection is on the line whose equation is
y = x +1,
the y coordinate will be
(0) + 1 = 1.
The reflection is then