Exercise 27 Page 221

Before we begin, let's assign names to the given lines for easier reference. To find the distance between $\ell$ and $k,$ we will follow a three-step process.

1. Pick a point on line $\ell$ and construct the perpendicular line $p$ through it.
2. Find the intersection point between lines $k$ and $p.$
3. Find the distance between the point chosen in the first step and the point found in the second step.

Finding the Equation of a Perpendicular Line

The slope of $\ell$ is $\frac{1}{4}.$ Because they are parallel, this is also the slope of $k.$ This implies that the slope of the perpendicular line $p$ must be $m=\text{-}4.$ As our point of intersection with line $\ell ,$ we will use the $y\text{-}$intercept, $(0,2).$ We can substitute these values into the point-slope form to write the equation of the line. This new equation, $y=\text{-}4x+2,$ is the equation of line $p.$

Finding the Intersection Point between Lines $k$ and $p$

To find the intersection point between lines $k$ and $p,$ we can create a system of equations. Let's find the solution using the Substitution Method. To find the $y\text{-}$coordinate, we will substitute $x=4$ into the second equation. The point of intersection is $(4,\text{-}14)$ which we will call point $P.$

Finding the Distance between the Two Points

Finally, to find the distance between $\ell$ and $k,$ we must find the distance between the point on $\ell ,$ $(0,2),$ and the point on $k,$ $P(4,\text{-}14).$ The distance between the lines is $4\sqrt{17}.$