Question

a city planner is rerouting traffic in order to work on a stretch of road. the equation of the path of the old route can be described as $y = \frac{2}{5}x - 4$. what should the equation of the new route be if it is to be perpendicular to the old route and will go through point (q, p)? (4 points)\
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$\bigcirc$ $y - q = -\frac{5}{2}(x - p)$\
\
$\bigcirc$ $y - q = \frac{2}{5}(x - p)$\
\
$\bigcirc$ $y - p = -\frac{5}{2}(x - q)$\
\
$\bigcirc$ $y - p = \frac{2}{5}(x - q)$

Explanation:

Step1: Find slope of old route

The old route is \( y = \frac{2}{5}x - 4 \), so its slope \( m_1=\frac{2}{5} \).

Step2: Find slope of new route

For perpendicular lines, \( m_1\times m_2=-1 \). So \( \frac{2}{5}\times m_2 = -1 \), solving gives \( m_2 = -\frac{5}{2} \).

Step3: Use point - slope form

The new route passes through \( (Q, P) \). Point - slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(Q,P) \) and \( m = -\frac{5}{2} \). So the equation is \( y - P=-\frac{5}{2}(x - Q) \).

Answer:

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