To determine the equation of the new route, we need to follow these steps:
1. Identify the slope of the old route: The old route is given by the equation [tex]\(y = \frac{2}{5} x - 4\)[/tex]. The slope of this line is the coefficient of [tex]\(x\)[/tex], which is [tex]\(\frac{2}{5}\)[/tex].
2. Find the slope of the new route: Since the new route is to be perpendicular to the old route, its slope will be the negative reciprocal of the old route's slope. The negative reciprocal of [tex]\(\frac{2}{5}\)[/tex] is [tex]\(-\frac{5}{2}\)[/tex].
3. Form the equation of the perpendicular line: The point-slope form of the equation of a line is [tex]\(y - Q = m(x - P)\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\((P, Q)\)[/tex] is the given point through which the line passes.
4. Substitute the slope and point into the point-slope form: We substitute the slope [tex]\(-\frac{5}{2}\)[/tex] and the point [tex]\((P, Q)\)[/tex] into the equation. This gives us:
[tex]\[
y - Q = -\frac{5}{2}(x - P)
\][/tex]
After these steps, the equation for the new route is:
[tex]\[
y - Q = -\frac{5}{2}(x - P)
\][/tex]
Hence, the correct option is:
[tex]\[
\boxed{y - Q = -\frac{5}{2}(x - P)}
\][/tex]
This corresponds to the choice [tex]\(1\)[/tex].
