Question
City planners are making plans for new roads. One road is shown as a line on the graph, and the second road will be perpendicular to the first, and will pass through the point (-2,4).
Graph the second line by first clicking on
(-2,4), then clicking on a point that would create a line perpendicular to the given line.
Graph the second line by first clicking on
(-2,4), then clicking on a point that would create a line perpendicular to the given line.
Answers
Answered by
GPT-4o mini
To graph the second line that is perpendicular to the given line and passes through the point (-2, 4), we first need to determine the slope of the original line.
1. **Identify the slope of the original line**: If you have the equation of the first line, you can find the slope (let's call it \( m_1 \)). For a line in the slope-intercept form \( y = mx + b \), the slope \( m \) can be easily identified.
2. **Calculate the negative reciprocal of the slope**: The slope of the line that is perpendicular to this one (let's call it \( m_2 \)) is the negative reciprocal of \( m_1 \). This means:
\[
m_2 = -\frac{1}{m_1}
\]
3. **Choose a point for the second line**: Since we know it passes through (-2, 4), we can find another point on this line.
4. **Use the point-slope form of the line equation**: Using the slope \( m_2 \) and the point (-2, 4), you can use the point-slope form to find another point.
If \( m_2 \) is your new slope, you can find a second point by moving right (increasing \( x \)) by 1 unit (for example), and then calculating the new \( y \) value:
\[
y - y_1 = m_2 (x - x_1)
\]
Substituting \( x_1 = -2 \) and \( y_1 = 4 \):
\[
y - 4 = m_2 (x + 2)
\]
Now, for instance, if you plug in \( x = -1 \) (which is one unit to the right of -2):
\[
y - 4 = m_2 (-1 + 2) = m_2 (1)
\]
\[
y = 4 + m_2
\]
5. **Plot the points**: Once you have the points (-2, 4) and the point you calculated using \( m_2 \), you can plot both points on the graph and draw a line through them.
Finally, you can finish graphing your perpendicular line based on these points. If you do not have the slope of the first line, the approach still works; simply choose an appropriate slope for your perpendicular line that might fit within the context of your project or the graph scale.
1. **Identify the slope of the original line**: If you have the equation of the first line, you can find the slope (let's call it \( m_1 \)). For a line in the slope-intercept form \( y = mx + b \), the slope \( m \) can be easily identified.
2. **Calculate the negative reciprocal of the slope**: The slope of the line that is perpendicular to this one (let's call it \( m_2 \)) is the negative reciprocal of \( m_1 \). This means:
\[
m_2 = -\frac{1}{m_1}
\]
3. **Choose a point for the second line**: Since we know it passes through (-2, 4), we can find another point on this line.
4. **Use the point-slope form of the line equation**: Using the slope \( m_2 \) and the point (-2, 4), you can use the point-slope form to find another point.
If \( m_2 \) is your new slope, you can find a second point by moving right (increasing \( x \)) by 1 unit (for example), and then calculating the new \( y \) value:
\[
y - y_1 = m_2 (x - x_1)
\]
Substituting \( x_1 = -2 \) and \( y_1 = 4 \):
\[
y - 4 = m_2 (x + 2)
\]
Now, for instance, if you plug in \( x = -1 \) (which is one unit to the right of -2):
\[
y - 4 = m_2 (-1 + 2) = m_2 (1)
\]
\[
y = 4 + m_2
\]
5. **Plot the points**: Once you have the points (-2, 4) and the point you calculated using \( m_2 \), you can plot both points on the graph and draw a line through them.
Finally, you can finish graphing your perpendicular line based on these points. If you do not have the slope of the first line, the approach still works; simply choose an appropriate slope for your perpendicular line that might fit within the context of your project or the graph scale.