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.
a. choose the correct answer below.
\\(\bigcirc\\) a. they intersect at exactly one point.
\\(\bigcirc\\) b. their y - intercepts have opposite signs.
\\(\bigcirc\\) c. the product of their slopes is 1
\\(\bigcirc\\) d. the product of their slopes is - 1.

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 perpendicularity condition

Calculate product of slopes: $m_1 \times m_2 = \frac{1}{5} \times (-5) = -1$.

Step3: Analyze graph viewing rectangle

In $[-10,10,1]$ by $[-10,10,1]$, the horizontal/vertical scales differ, so lines may not look perpendicular visually.

Step4: Explain zoom square feature

Zoom square makes scales equal, so lines appear perpendicular, matching the mathematical condition.

Answer:

a. D. The product of their slopes is - 1
b. No, the lines will not appear perpendicular in this viewing rectangle because the horizontal and vertical axis scales are not equal, distorting the angle between them.
c. After using the zoom square feature, the lines will appear perpendicular. This happens because the zoom square feature sets equal horizontal and vertical scales, which correctly represents the 90-degree angle between the mathematically perpendicular lines.