Question

If \( PS \) is the median of the triangle with vertices \( P(2, 2) \), \( Q(6, -1) \), and \( R(7, 3) \), then the equation of the line passing through \( (1, -1) \) and parallel to \( PS \) is:

• A. \( 4x + 7y + 3 = 0 \)
• B. \( 2x - 9y - 11 = 0 \)
• C. \( 4x - 7y - 11 = 0 \)
• D. \( 2x + 9y + 7 = 0 \)

Hint:

To find the line parallel to a segment, compute the slope of the segment and then use point-slope form with the new point. Don’t forget to simplify.

Answer 1

The Correct Option is D

We are given a triangle with vertices: \[ P = (2, 2), \quad Q = (6, -1), \quad R = (7, 3) \] We are told that \( PS \) is the **median**, which means \( S \) is the midpoint of side \( QR \). First, compute the coordinates of midpoint \( S \): \[ S = \left( \frac{6 + 7}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{13}{2}, 1 \right) \] So, line \( PS \) passes through: \[ P(2, 2), \quad S\left( \frac{13}{2}, 1 \right) \] Now, compute the slope of \( PS \): \[ m = \frac{1 - 2}{\frac{13}{2} - 2} = \frac{-1}{\frac{9}{2}} = -\frac{2}{9} \] Thus, any line parallel to \( PS \) will also have slope \( m = -\frac{2}{9} \). We are to find the equation of the line: - Passing through point \( (1, -1) \) - With slope \( m = -\frac{2}{9} \) Use point-slope form: \[ y - (-1) = -\frac{2}{9}(x - 1) \Rightarrow y + 1 = -\frac{2}{9}x + \frac{2}{9} \Rightarrow 9y + 9 = -2x + 2 \Rightarrow 2x + 9y + 7 = 0 \] % Final Answer \[ \boxed{2x + 9y + 7 = 0} \]