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5 P(3. 4). Q(7. -2) and R(-2. -1) are the vertices of APQR. Write down the equation of the median of the triangle through R to QP. [The use of the set of axes below is optional.]

5 P(3. 4). Q(7. -2) and R(-2. -1) are the vertices of APQR. Write down the equation of the median of the triangle through R to QP. [The use of the set of axes below is optional.]

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 5:** Given the points P(3, 4), Q(7, -2), and R(-2, -1) as the vertices of triangle △PQR, derive the equation of the median of the triangle that passes through R to the midpoint of segment PQ. **Solution Explanation:** To determine the equation of the median, find the midpoint of segment PQ: The midpoint M of PQ is calculated as follows: M = \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\) Substitute the coordinates: M = \(\left( \frac{3 + 7}{2}, \frac{4 + (-2)}{2} \right)\) M = \(\left( \frac{10}{2}, \frac{2}{2} \right)\) M = (5, 1) Now, find the equation of the line (median) passing through points R(-2, -1) and M(5, 1). Use the slope formula: Slope \(m\) = \(\frac{y_2 - y_1}{x_2 - x_1}\) Substitute the coordinates of R and M: \(m\) = \(\frac{1 - (-1)}{5 - (-2)}\) \(m\) = \(\frac{2}{7}\) Use the point-slope form to find the equation: \(y - y_1 = m(x - x_1)\) Substitute \(m\), \(x_1 = -2\), and \(y_1 = -1\): \(y + 1 = \frac{2}{7}(x + 2)\) Simplify to get the equation in the slope-intercept form: \(y = \frac{2}{7}x + \frac{4}{7} - 1\) \(y = \frac{2}{7}x - \frac{3}{7}\) Thus, the equation of the median is: \(y = \frac{2}{7}x - \frac{3}{7}\) **Graph Explanation:** The grid indicates a coordinate plane that can be used to plot the points and draw the median line. This visual aid helps
Transcribed Image Text:**Problem 5:** Given the points P(3, 4), Q(7, -2), and R(-2, -1) as the vertices of triangle △PQR, derive the equation of the median of the triangle that passes through R to the midpoint of segment PQ. **Solution Explanation:** To determine the equation of the median, find the midpoint of segment PQ: The midpoint M of PQ is calculated as follows: M = \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\) Substitute the coordinates: M = \(\left( \frac{3 + 7}{2}, \frac{4 + (-2)}{2} \right)\) M = \(\left( \frac{10}{2}, \frac{2}{2} \right)\) M = (5, 1) Now, find the equation of the line (median) passing through points R(-2, -1) and M(5, 1). Use the slope formula: Slope \(m\) = \(\frac{y_2 - y_1}{x_2 - x_1}\) Substitute the coordinates of R and M: \(m\) = \(\frac{1 - (-1)}{5 - (-2)}\) \(m\) = \(\frac{2}{7}\) Use the point-slope form to find the equation: \(y - y_1 = m(x - x_1)\) Substitute \(m\), \(x_1 = -2\), and \(y_1 = -1\): \(y + 1 = \frac{2}{7}(x + 2)\) Simplify to get the equation in the slope-intercept form: \(y = \frac{2}{7}x + \frac{4}{7} - 1\) \(y = \frac{2}{7}x - \frac{3}{7}\) Thus, the equation of the median is: \(y = \frac{2}{7}x - \frac{3}{7}\) **Graph Explanation:** The grid indicates a coordinate plane that can be used to plot the points and draw the median line. This visual aid helps
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