To find the equation of a new route that is perpendicular to the old route, we first need to determine the slope of the old route. The old route is given by the equation:
\[ y = \frac{2}{5}x - 4 \]
From this equation, we can see that the slope (m) of the old route is \( \frac{2}{5} \).
To find the slope of the new route, which is perpendicular to the old route, we use the fact that the product of the slopes of two perpendicular lines is -1. If the slope of the old route is \( m_1 = \frac{2}{5} \), then the slope of the new route \( m_2 \) is:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{2}{5}} = -\frac{5}{2} \]
Now we have the slope of the new route as \( -\frac{5}{2} \).
We can then use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is a point on the line (in this case, it is point (P, Q)), and \( m \) is the slope. Substituting in \( m = -\frac{5}{2} \) and the point \( (P, Q) \), we have:
\[ y - Q = -\frac{5}{2}(x - P) \]
Thus, the correct equation for the new route, which is perpendicular to the old route and goes through the point (P, Q), is:
\[ y - Q = -\frac{5}{2}(x - P) \]
This corresponds to the first option provided:
y − Q = negative five halves(x − P)