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(Parallel and Perpendicular Lines HC)

The work of two students is shown for the problem: What is the slope of a line that is perpendicular to a line that passes through [tex]$(-22,13)$[/tex] and [tex]$(17,12)$[/tex]?

[tex]\[
\begin{tabular}{|c|c|}
\hline
Student A & Student B \\
\hline
$m=\frac{12-13}{17-(-22)}=-\frac{1}{39}$ & $m=\frac{12-13}{17-(-22)}=-\frac{1}{39}$ \\
\hline
The slope of the perpendicular line is -39. & The slope of the perpendicular line is 39. \\
\hline
\end{tabular}
\][/tex]

Part A: Which student has the correct slope of the perpendicular line? (2 points)

Part B: Explain how the answer to Part A is correct. (2 points)

Sagot :

GinnyAnsweridn
To determine the correct slope of a line that is perpendicular to the line passing through the points [tex]\((-22, 13)\)[/tex] and [tex]\((17, 12)\)[/tex], let's walk through the solution step-by-step:

### Part A: Identifying the Correct Slope
First, we need to calculate the slope of the line that passes through these two points. The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:

[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substituting the given coordinates [tex]\((-22, 13)\)[/tex] and [tex]\((17, 12)\)[/tex]:

[tex]\[ m = \frac{12 - 13}{17 - (-22)} = \frac{12 - 13}{17 + 22} = \frac{-1}{39} \][/tex]

So, the slope of the line passing through these points is [tex]\( -\frac{1}{39} \)[/tex].

Next, we need to determine the slope of a line that is perpendicular to this line. The slope of a perpendicular line is the negative reciprocal of the original slope. The negative reciprocal of [tex]\( m \)[/tex] is calculated as follows:

[tex]\[ \text{Perpendicular slope} = -\frac{1}{m} \][/tex]

Given [tex]\( m = -\frac{1}{39} \)[/tex]:

[tex]\[ \text{Perpendicular slope} = -\frac{1}{-\frac{1}{39}} = 39 \][/tex]

Thus, the correct slope of the line that is perpendicular to the line passing through [tex]\((-22, 13)\)[/tex] and [tex]\((17, 12)\)[/tex] is indeed [tex]\( 39 \)[/tex].

From the calculations, it is clear that Student B has the correct slope of the perpendicular line.

### Part B: Explanation of the Correct Answer
To verify that Student B's answer is correct, let's explain the reasoning:

1. Calculate the Original Slope:
- Using the slope formula for the points [tex]\((-22, 13)\)[/tex] and [tex]\((17, 12)\)[/tex], we obtained:

[tex]\[ m = \frac{-1}{39} \][/tex]

2. Find the Slope of the Perpendicular Line:
- The negative reciprocal of [tex]\( -\frac{1}{39} \)[/tex] is [tex]\( 39 \)[/tex].
- This calculation is based on the property of perpendicular lines: if two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other.

Therefore, Student B correctly applied the concept of negative reciprocals to find the slope of the line that is perpendicular to the original line. Hence, the correct slope is [tex]\( 39 \)[/tex], and Student B's solution is accurate.
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