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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 [tex]y=\frac{2}{5} x-4[/tex]. What should the equation of the new route be if it is to be perpendicular to the old route and will go through point [tex]\((Q, P)\)[/tex]?

A. [tex]y - Q = -\frac{5}{2}(x - P)[/tex]
B. [tex]y - Q = \frac{2}{5}(x - P)[/tex]
C. [tex]y - P = -\frac{5}{2}(x - Q)[/tex]
D. [tex]y - P = \frac{2}{5}(x - Q)[/tex]

Sagot :

GinnyAnsweridn
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]
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