To solve this problem, let's go through the steps of finding the equation of the new route.
1. Identify the slope of the old route:
The equation of the old route is given as [tex]\( y = \frac{2}{5} x - 4 \)[/tex]. The slope (m) of this line is [tex]\(\frac{2}{5}\)[/tex].
2. Find the slope of the new route:
The new route is perpendicular to the old route. For two lines to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]. Let the slope of the new route be [tex]\(m_{\text{new}}\)[/tex].
[tex]\[
m \times m_{\text{new}} = -1 \implies \left(\frac{2}{5}\right) \times m_{\text{new}} = -1
\][/tex]
Solving for [tex]\(m_{\text{new}}\)[/tex]:
[tex]\[
m_{\text{new}} = -\frac{1}{m} = -\frac{1}{\left(\frac{2}{5}\right)} = -\frac{5}{2}
\][/tex]
Hence, the slope of the new route is [tex]\(-\frac{5}{2}\)[/tex].
3. Point-Slope Form of the new route:
The new route passes through the point [tex]\((Q, P)\)[/tex] which is [tex]\((1, 4)\)[/tex].
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting the slope [tex]\(m_{\text{new}} = -\frac{5}{2}\)[/tex] and the point [tex]\((x_1, y_1) = (1, 4)\)[/tex]:
[tex]\[
y - 4 = -\frac{5}{2}(x - 1)
\][/tex]
Thus, the equation of the new route is:
[tex]\[
y - 4 = -\frac{5}{2}(x - 1)
\][/tex]
Therefore, the correct choice among the given options is:
[tex]\[y - P = -\frac{5}{2}(x - Q).\][/tex]
