Question

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. the city 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 (1,1), (3,4)

Answer 1

To determine the slope of the new road that is perpendicular to Main Street, we first need to find the slope of Main Street based on the given points.

The two points provided are (1, 1) and (3, 4). The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

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

Plugging in the points (1, 1) and (3, 4):

- \( x_1 = 1 \)
- \( y_1 = 1 \)
- \( x_2 = 3 \)
- \( y_2 = 4 \)

So the slope \( m \) becomes:

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

To find the slope of a line that is perpendicular to this line, we take the negative reciprocal of the slope. The negative reciprocal of \( \frac{3}{2} \) is:

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

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

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