To find the equation of the line passing through [tex]\((1, -1)\)[/tex] and parallel to the median [tex]\(PS\)[/tex] of the triangle with vertices [tex]\(P(2, 2)\)[/tex], [tex]\(Q(6, -1)\)[/tex], and [tex]\(R(7, 3)\)[/tex], we proceed with the following steps:
### Step 1: Find the midpoint of [tex]\(QR\)[/tex]
First, we calculate the coordinates of the midpoint [tex]\(S\)[/tex] of side [tex]\(QR\)[/tex]:
[tex]\[
S = \left( \frac{6+7}{2}, \frac{-1+3}{2} \right) = \left( \frac{13}{2}, \frac{2}{2} \right) = (6.5, 1)
\][/tex]
### Step 2: Calculate the slope of [tex]\(PS\)[/tex]
Next, we find the slope of the median [tex]\(PS\)[/tex]:
[tex]\[
\text{slope of } PS = \frac{S_y - P_y}{S_x - P_x} = \frac{1 - 2}{6.5 - 2} = \frac{-1}{4.5}
\][/tex]
### Step 3: Use the point-slope form to find the line passing through [tex]\((1, -1)\)[/tex]
We need the equation of the line passing through [tex]\((1, -1)\)[/tex] and parallel to [tex]\(PS\)[/tex]. Since the lines are parallel, they share the same slope. Using the point-slope form of the equation of a line:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
substituting [tex]\(m = -\frac{1}{4.5}\)[/tex], [tex]\(x_1 = 1\)[/tex], and [tex]\(y_1 = -1\)[/tex], we get:
[tex]\[
y + 1 = -\frac{1}{4.5}(x - 1)
\][/tex]
Simplifying this:
[tex]\[
y + 1 = -\frac{1}{4.5}x + \frac{1}{4.5}
\][/tex]
To clear the fraction, multiply everything by 4.5:
[tex]\[
4.5y + 4.5 = -x + 1
\][/tex]
Rearrange to standard form [tex]\(Ax + By + C = 0\)[/tex]:
[tex]\[
x + 4.5y + 3.5 = 0
\][/tex]
### Step 4: Convert to integer coefficients
To match the provided options, we multiply all terms by 2 to avoid fractional coefficients:
[tex]\[
2x + 9y + 7 = 0
\][/tex]
Thus, the equation of the line passing through [tex]\((1, -1)\)[/tex] and parallel to [tex]\(PS\)[/tex] is:
[tex]\[
\boxed{2x + 9y + 7 = 0}
\][/tex]
Therefore, the correct answer is:
d. [tex]\(2 x + 9 y + 7 = 0\)[/tex]
