To find the equation of the new route that is perpendicular to the old route, we need to follow these steps:

1. Find the slope of the old route.

The given equation of the old route is:

$y = \frac{2}{5}x - 4$

This is in the slope-intercept form $y = mx + b$, where $m$ represents the slope.

From the equation, the slope of the old route is $m = \frac{2}{5}$.

2. Find the slope of the new route.

The slope of two perpendicular lines is the negative reciprocal of each other. To find the slope of the new route, we take the negative reciprocal of $\frac{2}{5}$:

$\text{slope of the new route} = -\frac{5}{2}$

3. Use the point-slope form to write the equation of the new route.

The point-slope form of the equation of a line is:

$y - Q = m(x - P)$

Where $m$ is the slope, and $(P, Q)$ is the point the line passes through.

We already know the slope of the new route is $-\frac{5}{2}$, and the line goes through the point $(P, Q)$. So, the equation becomes:

$y - Q = -\frac{5}{2}(x - P)$

4. Check the answer choices.

This matches the first answer choice:

$y - Q = -\frac{5}{2}(x - P)$

Conclusion:

The correct equation for the new route is:

$y - Q = -\frac{5}{2}(x - P)$

Let me know if you would like more details on this or if you have any questions!

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Here are 5 related questions to deepen your understanding:

1. How do you calculate the slope of a line from an equation in slope-intercept form?
2. How does the concept of perpendicular lines relate to their slopes?
3. Can the equation of a line ever have two slopes? How does this affect perpendicular lines?
4. What is the general form of the equation of a line, and how can it be converted into slope-intercept form?
5. How would the equation change if the point $(P, Q)$ was given as $(2, 3)$?

Tip: To quickly determine if two lines are perpendicular, simply multiply their slopes. If the product is $-1$, the lines are perpendicular.