At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To find the equation of the line that passes through the midpoint of the line segment joining the points \((4, 3)\) and \((2, 5)\) and is perpendicular to the line \(2x + 5y + 3 = 0\), follow these steps:
1. Find the Midpoint of the Line Segment:
The midpoint \((x_m, y_m)\) of the line segment joining the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
[tex]\[ x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2} \][/tex]
For the points \((4, 3)\) and \((2, 5)\):
[tex]\[ x_m = \frac{4 + 2}{2} = 3, \quad y_m = \frac{3 + 5}{2} = 4 \][/tex]
So, the midpoint is \((3, 4)\).
2. Find the Slope of the Given Line:
The equation of the given line is \(2x + 5y + 3 = 0\). The general form of a linear equation is \(Ax + By + C = 0\), and the slope \(m\) of the line can be found using:
[tex]\[ m = -\frac{A}{B} \][/tex]
Here, \(A = 2\) and \(B = 5\), so the slope of the given line is:
[tex]\[ m = -\frac{2}{5} \][/tex]
3. Find the Slope of the Perpendicular Line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. So, if the slope of the original line is \(-\frac{2}{5}\), the slope \(m'\) of the perpendicular line is:
[tex]\[ m' = -\left(-\frac{5}{2}\right) = \frac{5}{2} \][/tex]
4. Write the Equation of the Perpendicular Line:
The equation of a line in slope-intercept form is given by:
[tex]\[ y = mx + c \][/tex]
We need to find the value of \(c\), the y-intercept. For a line passing through the point \((x_m, y_m) = (3, 4)\) with slope \(m' = \frac{5}{2}\), we have:
[tex]\[ 4 = \left(\frac{5}{2}\right) \cdot 3 + c \][/tex]
Now, solve for \(c\):
[tex]\[ 4 = \frac{15}{2} + c \\ 4 - \frac{15}{2} = c \\ c = \frac{8}{2} - \frac{15}{2} = -\frac{7}{2} = -3.5 \][/tex]
Therefore, the y-intercept \(c\) is \(-3.5\).
5. Combine the Slope and Y-Intercept into the Final Equation:
So, the equation of the line passing through \((3, 4)\) and perpendicular to \(2x + 5y + 3 = 0\) is:
[tex]\[ y = \frac{5}{2}x - 3.5 \][/tex]
Summarizing, the equation of the line is:
[tex]\[ y = \frac{5}{2}x - 3.5 \][/tex]
1. Find the Midpoint of the Line Segment:
The midpoint \((x_m, y_m)\) of the line segment joining the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
[tex]\[ x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2} \][/tex]
For the points \((4, 3)\) and \((2, 5)\):
[tex]\[ x_m = \frac{4 + 2}{2} = 3, \quad y_m = \frac{3 + 5}{2} = 4 \][/tex]
So, the midpoint is \((3, 4)\).
2. Find the Slope of the Given Line:
The equation of the given line is \(2x + 5y + 3 = 0\). The general form of a linear equation is \(Ax + By + C = 0\), and the slope \(m\) of the line can be found using:
[tex]\[ m = -\frac{A}{B} \][/tex]
Here, \(A = 2\) and \(B = 5\), so the slope of the given line is:
[tex]\[ m = -\frac{2}{5} \][/tex]
3. Find the Slope of the Perpendicular Line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. So, if the slope of the original line is \(-\frac{2}{5}\), the slope \(m'\) of the perpendicular line is:
[tex]\[ m' = -\left(-\frac{5}{2}\right) = \frac{5}{2} \][/tex]
4. Write the Equation of the Perpendicular Line:
The equation of a line in slope-intercept form is given by:
[tex]\[ y = mx + c \][/tex]
We need to find the value of \(c\), the y-intercept. For a line passing through the point \((x_m, y_m) = (3, 4)\) with slope \(m' = \frac{5}{2}\), we have:
[tex]\[ 4 = \left(\frac{5}{2}\right) \cdot 3 + c \][/tex]
Now, solve for \(c\):
[tex]\[ 4 = \frac{15}{2} + c \\ 4 - \frac{15}{2} = c \\ c = \frac{8}{2} - \frac{15}{2} = -\frac{7}{2} = -3.5 \][/tex]
Therefore, the y-intercept \(c\) is \(-3.5\).
5. Combine the Slope and Y-Intercept into the Final Equation:
So, the equation of the line passing through \((3, 4)\) and perpendicular to \(2x + 5y + 3 = 0\) is:
[tex]\[ y = \frac{5}{2}x - 3.5 \][/tex]
Summarizing, the equation of the line is:
[tex]\[ y = \frac{5}{2}x - 3.5 \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
