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Step 1: Identify the mid-point of QR
The coordinates of the vertices are:
P(2, 2), Q(6, –1), and R(7, 3).
Since PS is the median, S is the midpoint of QR. The midpoint S is given by:
$S = \left(\dfrac{x_Q + x_R}{2}, \dfrac{y_Q + y_R}{2}\right)$
Substituting the values:
$S = \left(\dfrac{6 + 7}{2}, \dfrac{-1 + 3}{2}\right) = \left(\dfrac{13}{2}, 1\right)$
Step 2: Find the slope of PS
The point P is (2, 2) and the point S is
$\left(\dfrac{13}{2}, 1\right)$. The slope of the line PS is given by:
$\text{slope} = \dfrac{y_P - y_S}{x_P - x_S}
= \dfrac{2 - 1}{2 - \dfrac{13}{2}}
= \dfrac{1}{2 - \dfrac{13}{2}}
= \dfrac{1}{\dfrac{4 - 13}{2}}
= \dfrac{1}{-\dfrac{9}{2}}
= -\dfrac{2}{9}$
Step 3: Use the slope and the given point to form the required line
We want the line passing through (1, –1) and parallel to PS. Parallel lines have the same slope, so the slope is $-\dfrac{2}{9}$. The point-slope form of a line is:
$y - y_1 = m (x - x_1)$
Substituting $(x_1, y_1) = (1, -1)$ and $m = -\dfrac{2}{9}$:
$y - (-1) = -\dfrac{2}{9} \left(x - 1\right)$
$y + 1 = -\dfrac{2}{9}x + \dfrac{2}{9}$
Step 4: Simplify to get the standard form
Multiply both sides by 9 to clear the fraction:
$9(y + 1) = 9\left(-\dfrac{2}{9}x + \dfrac{2}{9}\right)$
$9y + 9 = -2x + 2$
Bring all terms to one side:
$2x + 9y + 7 = 0$
Final Answer
The equation of the required line is:
$2x + 9y + 7 = 0$
