Question

Use the image to answer the question.

a coordinate plane with the x axis ranging from negative 5 to 5 and the y axis ranging from negative 5 to 5

Image Long DescriptionA point labeled gas station lies at left parenthesis 1 comma 1 right parenthesis. A point labeled post office lies at left parenthesis 3 comma 4 right parenthesis. A dotted-dashed line labeled main street runs through both points. Two dotted lines intersect at left parenthesis 3 comma zero right parenthesis. Two solid lines intersect at left parenthesis 1 comma 2 right parenthesis.

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 gas station lies at the point (1, 1)
, and the post office lies at the point (3, 4)
. The line that passes through both of these points represents Main Street. 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 point)
The slope of the new road must be
.

Answer 1

To find the slope of the new road that is perpendicular to Main Street, we first need to determine the slope of Main Street itself, which passes through the points (1, 1) and (3, 4).

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

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

For the points (1, 1) and (3, 4):

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

Now substituting these values into the slope formula:

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

Now that we have the slope of Main Street as \( \frac{3}{2} \), we can determine the slope of the road that is perpendicular to it. The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope.

To find the negative reciprocal of \( \frac{3}{2} \):

1. Take the reciprocal: the reciprocal of \( \frac{3}{2} \) is \( \frac{2}{3} \).
2. Change the sign: the negative reciprocal is \( -\frac{2}{3} \).

Therefore, the slope of the new road must be

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