Delaney is a city engineer and needs to create a scale map of the city. She has graphed several landmarks on Main Street, including the gas station and the post office is drawing up plans to construct a road perpendicular to Main Street. What must be the slope of the new road in order to prove it is perpendicular to main Street? express your answer as a whole number or fraction. Gas station is at (1,1) and post office at (4,3)

Answer 1

To determine the slope of the new road that is perpendicular to Main Street, we first need to calculate the slope of Main Street, which is the line segment connecting the gas station at (1, 1) and the post office at (4, 3).

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the coordinates of the gas station (1, 1) as \((x_1, y_1)\) and the post office (4, 3) as \((x_2, y_2)\):

\[ m = \frac{3 - 1}{4 - 1} = \frac{2}{3} \]

Now that we have the slope of Main Street, which is \( \frac{2}{3} \), we can determine the slope of the new road that will be perpendicular to it. The slopes of two perpendicular lines are negative reciprocals of each other.

To find the negative reciprocal of the slope \( \frac{2}{3} \), we take the reciprocal and change the sign:

\[ \text{Slope of the new road} = -\frac{1}{\left(\frac{2}{3}\right)} = -\frac{3}{2} \]

Thus, the slope of the new road that is perpendicular to Main Street must be:

\[ \boxed{-\frac{3}{2}} \]