Question

The vertices of \( \triangle PQR \) are \( P(2,1) \), \( Q(-2,3) \), and \( R(4,5) \). Find the equation of the median through the vertex \( R \).

Hint:

The median of a triangle connects a vertex to the midpoint of the opposite side. To find its equation, first find the midpoint of the opposite side, then use the point-slope formula.

Answer 1

Step 1: Find the midpoint of side \( PQ \).
The midpoint \( M \) of the line segment joining points \( P(2, 1) \) and \( Q(-2, 3) \) is: \[ M = \left( \frac{2 + (-2)}{2}, \frac{1 + 3}{2} \right) = (0, 2) \]
Step 2: Find the slope of the median.}
The median is the line joining vertex \( R(4, 5) \) to the midpoint \( M(0, 2) \). The slope \( m_{\text{median}} \) is: \[ m_{\text{median}} = \frac{5 - 2}{4 - 0} = \frac{3}{4} \]
Step 3: Use the point-slope form to find the equation of the median.}
Using the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] Substitute \( m = \frac{3}{4} \) and the point \( (0, 2) \): \[ y - 2 = \frac{3}{4}(x - 0) \] Simplify the equation: \[ y - 2 = \frac{3}{4}x \] \[ y = \frac{3}{4}x + 2 \] So, the equation of the median is \( y = \frac{3}{4}x + 2 \).