Chapter 4, Problem 15CR

To determine

if there is a parallel or perpendicular line

Answer to Problem 15CR

The lines $a,b$ have the same slope and different $y-intercepts$ , thus they are parallel. The lines are not perpendicular.

Explanation of Solution

Given:

Line a passes through (0, 4) and (4, 3)

Line b passes through (0, 1) and (4, 0)

Line c passes through (2, 0) and (4, 4)

Calculation:

Find the equation of line a:

  $\begin{array}{l}{m}_a=\frac{3-4}{4-0}=-0.25\\ y-4=m_a\left(x-0\right)\\ y-4=-0.25x\\ y=-0.25x+4\end{array}$

Find the equation of line b:

  $\begin{array}{l}{m}_b=\frac{0-1}{4-0}=-0.25\\ y-1=m_b\left(x-0\right)\\ y-4=-0.25x\\ y=-0.25x+1\end{array}$

find the equation of line c:

  $\begin{array}{l}{m}_c=\frac{4-0}{4-2}=2\\ y-0=m_c\left(x-2\right)\\ y=2x-4\end{array}$

The lines $a,b$ have the same slope and different $y-intercepts$ , thus they are parallel but do not coincide.

Find the slope of c:

  $m_c=2$

Check if the lines are perpendicular.

  $m_c\cdot m_a=2\left(-0.25\right)=-0.5\cancel{=}-1$

The lines are not perpendicular.

Conclusion:

The lines $a,b$ have the same slope and different $y-intercepts$ , thus they are parallel. The lines are not perpendicular.