Question

Alec is marking the edges of his garden.





One side of his garden has ends at (2,3) and (7,4). The next side of the garden will be perpendicular to this side. What is the slope of the next side?

Answer 1

To determine the slope of the next side of Alec's garden, which is perpendicular to the first side, we first need to find the slope of the line segment formed by the points (2, 3) and (7, 4).

The slope \( m \) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

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

For the points (2, 3) and (7, 4), we have:

- \( x_1 = 2 \)
- \( y_1 = 3 \)
- \( x_2 = 7 \)
- \( y_2 = 4 \)

Plugging these values into the slope formula:

\[
m = \frac{4 - 3}{7 - 2} = \frac{1}{5}
\]

The slope of the first side of the garden is \( \frac{1}{5} \).

Now, 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 \( \frac{1}{5} \):

1. Take the reciprocal: \( \frac{1}{\frac{1}{5}} = 5 \)
2. Change the sign: \( -5 \)

Thus, the slope of the next side of Alec's garden is:

\[
\boxed{-5}
\]