Question

a. why are the lines whose equations are $y = \frac{1}{5}x + 6$ and $y = -5x - 3$ perpendicular?
b. use a graphing utility to graph the equations in a $-10,10,1$ by $-10,10,1$ viewing rectangle. do the lines appear to be perpendicular?
c. now use the zoom square feature of your utility. describe what happens to the graphs. explain why this is so.
c. choose the correct answer below.
\\(\bigcirc\\) a. nothing changes because the functions themselves have not changed.
\\(\bigcirc\\) b. the lines now appear to be perpendicular because in this new viewing window, the distances between tick marks on the x- and y-axes are equal.
\\(\bigcirc\\) c. the lines now appear to be less perpendicular than before because this new viewing window shows more of the two graphs.

Explanation:

Step1: Identify slopes of lines

For $y=\frac{1}{5}x+6$, slope $m_1=\frac{1}{5}$; for $y=-5x-3$, slope $m_2=-5$.

Step2: Check perpendicular slope rule

Multiply slopes: $m_1 \times m_2 = \frac{1}{5} \times (-5) = -1$

Step3: Analyze non-square viewing window

In $[-10,10,1]$ by $[-10,10,1]$, axis scales match visually, so lines appear perpendicular.

Step4: Explain zoom square feature

Zoom square makes x/y tick mark distances equal, correcting any visual distortion.

Answer:

a. The product of their slopes is $-1$: $\frac{1}{5} \times (-5) = -1$, which is the condition for two lines to be perpendicular.
b. Yes, the lines appear to be perpendicular in this viewing rectangle because the scale of the x-axis and y-axis is the same.
c. B. The lines now appear to be perpendicular because in this new viewing window, the distances between tick marks on the x- and y-axes are equal.