To find the slope of the new road that must be perpendicular to Main Street, we first need to calculate the slope of Main Street between the gas station and the post office.
The coordinates of the gas station are \((4, 3)\) and the coordinates of the post office are \((1, 1)\).
The formula for the slope (\(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Plugging in the coordinates of the gas station \((4, 3)\) as \((x_1, y_1)\) and the post office \((1, 1)\) as \((x_2, y_2)\):
\[ m = \frac{1 - 3}{1 - 4} = \frac{-2}{-3} = \frac{2}{3} \]
So, the slope of Main Street is \(\frac{2}{3}\).
For two lines to be perpendicular, the product of their slopes must equal \(-1\). Let \(m_1\) be the slope of Main Street (which we found to be \(\frac{2}{3}\)) and \(m_2\) be the slope of the new road. Then:
\[ m_1 \cdot m_2 = -1 \]
Substituting \(m_1 = \frac{2}{3}\):
\[ \frac{2}{3} \cdot m_2 = -1 \]
To find \(m_2\):
\[ m_2 = -1 \div \frac{2}{3} = -1 \cdot \frac{3}{2} = -\frac{3}{2} \]
Thus, the slope of the new road that is perpendicular to Main Street must be \(-\frac{3}{2}\).
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