Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
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 hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
