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Chapter 14: Problem 1347

Let PS be the median of the triangle with vertices \(\mathrm{P}(2,2)\), \(\mathrm{Q}(6,-1)\) and \(\mathrm{R}(7,3)\). The equation of the line passing through \((1,-1)\) and parallel to \(P S\) is (a) \(2 x-9 y-7=0\) (b) \(2 x-9 y-11=0\) (c) \(2 x+9 y-11=0\) (d) \(2 x+9 y+7=0\)

Short Answer

Expert verified
The short answer is: the equation of the line parallel to the median PS and passing through the point (1, -1) is (b) \(2x - 9y - 11 = 0\).

Step by step solution

01

Find the midpoint of QR.

The coordinates of points Q and R are given as (6,-1) and (7,3) respectively. We can find the midpoint S of QR using the midpoint formula: \[ S = \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) = \left(\frac{6+7}{2}, \frac{-1+3}{2}\right) = (6.5, 1) \]
02

Calculate the slope of PS.

Now that we have the coordinates of the endpoints of the median PS, i.e., P(2,2) and S(6.5,1), we can find the slope of the line segment PS using the following formula: \[ m_{PS} =\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{1-2}{6.5-2} = \frac{-1}{4.5} = -\frac{2}{9} \]
03

Write the equation of a line parallel to PS and passing through (1,-1).

We have the slope of the line we need, \(-\frac{2}{9}\). Since this line passes through the point (1,-1), we can use the point-slope form for the equation of the line: \[ y - y_1 = m(x - x_1) \] Substituting the given point and the slope of the parallel line, we get: \[ y - (-1) = -\frac{2}{9}(x - 1) \] Simplifying the equation, we have: \[ 9(y + 1) = -2(x - 1) \] Expanding and rearranging this equation, we get the final equation as: \[ 2x - 9y - 11 = 0 \] So, the correct answer is (b) \(2x - 9y - 11 = 0\).

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