To find which sides of the given four-sided figure have negative slopes, we need to evaluate the slopes of each pair of endpoints forming the sides.
Let's analyze each set of points provided and determine the pairs that form the sides of the figure with negative slopes.
Given points:
1. [tex]\((1, -1)\)[/tex], [tex]\((2, -4)\)[/tex]
2. [tex]\((1, 4)\)[/tex], [tex]\((2, 1)\)[/tex]
3. [tex]\((5, 1)\)[/tex], [tex]\((4, 4)\)[/tex]
These points are analyzed in pairs to determine the slope ([tex]\(m\)[/tex]) formed by the line segment connecting them. The slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Next, let's review the results:
1. For the pair [tex]\((1, -1)\)[/tex] and [tex]\((2, -4)\)[/tex]:
[tex]\[
m = \frac{-4 - (-1)}{2 - 1} = \frac{-4 + 1}{2 - 1} = \frac{-3}{1} = -3
\][/tex]
The slope is [tex]\(-3\)[/tex], which is negative.
2. For the pair [tex]\((1, 4)\)[/tex] and [tex]\((2, 1)\)[/tex]:
[tex]\[
m = \frac{1 - 4}{2 - 1} = \frac{1 - 4}{1} = \frac{-3}{1} = -3
\][/tex]
The slope is [tex]\(-3\)[/tex], which is negative.
3. For the pair [tex]\((5, 1)\)[/tex] and [tex]\((4, 4)\)[/tex]:
[tex]\[
m = \frac{4 - 1}{4 - 5} = \frac{4 - 1}{-1} = \frac{3}{-1} = -3
\][/tex]
The slope is [tex]\(-3\)[/tex], which is negative.
Therefore, the sides of the four-sided figure that have negative slopes are given by the following endpoints:
- [tex]\((1, -1)\)[/tex] and [tex]\((2, -4)\)[/tex]
- [tex]\((1, 4)\)[/tex] and [tex]\((2, 1)\)[/tex]
- [tex]\((5, 1)\)[/tex] and [tex]\((4, 4)\)[/tex]
Thus, the endpoints of the sides of the figure with negative slopes are:
[tex]\[
[((1, -1), (2, -4)), ((1, 4), (2, 1)), ((5, 1), (4, 4))]
\][/tex]
