Graph a line that is perpendicular to the given line. Determine the slope of the given line and the one you graphed in simplest form.
Understand the Problem
The question is asking to graph a line that is perpendicular to a given line and to determine the slope of both the given line and the newly graphed line in simplest form.
Answer
The slope of the given line is $m = -1$, and the slope of the perpendicular line is $m_{\perp} = 1$.
Answer for screen readers
The slope of the given line is $m = -1$, and the slope of the perpendicular line is $m_{\perp} = 1$.
Steps to Solve
- Identify the slope of the given line
To find the slope of the given line, we need to use its endpoints or its equation. From the graph, we can see that the line has a negative slope. If we select two points on this line, say (0, 3) and (5, -2), we can calculate the slope using the formula:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Substituting the coordinates into the formula gives:
$$ m = \frac{-2 - 3}{5 - 0} = \frac{-5}{5} = -1 $$
So, the slope of the given line is $m = -1$.
- Determine the slope of the perpendicular line
The slope of a line that is perpendicular to another is the negative reciprocal of the original line's slope. Since the slope of the given line is $-1$, the perpendicular slope $m_{\perp}$ is:
$$ m_{\perp} = -\frac{1}{m} = -\frac{1}{-1} = 1 $$
Thus, the slope of the perpendicular line is $m_{\perp} = 1$.
- Graph the perpendicular line
To graph the line with slope $1$, start from a point on the original line. For example, we can use the point (0, 3). Using the slope, rise 1 unit and run 1 unit to the right. Plot another point at (1, 4). Connect these points to form the perpendicular line.
The slope of the given line is $m = -1$, and the slope of the perpendicular line is $m_{\perp} = 1$.
More Information
The concept of perpendicular lines is fundamental in geometry. Perpendicular lines intersect at right angles (90 degrees), and their slopes are related through the negative reciprocal relationship.
Tips
- Confusing the slope of the original line with the slope of the perpendicular line; it's important to remember to take the negative reciprocal.
- Miscalculating the slope when determining it from two points; always double-check the coordinates used in the slope formula.
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