Answer :
To determine which set of points forms a four-sided figure where exactly two of the sides have negative slopes, we carefully analyze the given point configurations. We will reference each configuration to identify the slopes between pairs of points and then count how many of these slopes are negative.
Let's review each given set of points:
### Option 1:
Points: [tex]\[ (-4, -4), (-4, -1), (-1, -4), (-1, -1) \][/tex]
- Slope between [tex]\((-4, -4)\)[/tex] and [tex]\((-4, -1)\)[/tex]: Vertical line, undefined slope.
- Slope between [tex]\((-4, -1)\)[/tex] and [tex]\((-1, -4)\)[/tex]: [tex]\(\frac{-4 + 1}{-1 + 4} = \frac{-3}{3} = -1\)[/tex]
- Slope between [tex]\((-1, -4)\)[/tex] and [tex]\((-1, -1)\)[/tex]: Vertical line, undefined slope.
- Slope between [tex]\((-1, -1)\)[/tex] and [tex]\((-4, -4)\)[/tex]: [tex]\(\frac{-4 + 1}{-4 + 1} = \frac{-3}{-3} = 1\)[/tex]
This configuration has 1 negative slope.
### Option 2:
Points: [tex]\[ (-2, -4), (-1, -1), (1, -1), (2, -4) \][/tex]
- Slope between [tex]\((-2, -4)\)[/tex] and [tex]\((-1, -1)\)[/tex]: [tex]\(\frac{-1 + 4}{-1 + 2} = \frac{3}{1} = 3\)[/tex]
- Slope between [tex]\((-1, -1)\)[/tex] and [tex]\((1, -1)\)[/tex]: [tex]\(\frac{-1 + 1}{1 + 1} = \frac{0}{2} = 0\)[/tex]
- Slope between [tex]\((1, -1)\)[/tex] and [tex]\((2, -4)\)[/tex]: [tex]\(\frac{-4 + 1}{2 - 1} = \frac{-3}{1} = -3\)[/tex]
- Slope between [tex]\((2, -4)\)[/tex] and [tex]\((-2, -4)\)[/tex]: Horizontal line, slope is 0.
This configuration has 1 negative slope.
### Option 3:
Points: [tex]\[ (1, 1), (2, 4), (5, 4), (4, 1) \][/tex]
- Slope between [tex]\((1, 1)\)[/tex] and [tex]\((2, 4)\)[/tex]: [tex]\(\frac{4 - 1}{2 - 1} = \frac{3}{1} = 3\)[/tex]
- Slope between [tex]\((2, 4)\)[/tex] and [tex]\((5, 4)\)[/tex]: Horizontal line, slope is 0.
- Slope between [tex]\((5, 4)\)[/tex] and [tex]\((4, 1)\)[/tex]: [tex]\(\frac{1 - 4}{4 - 5} = \frac{-3}{-1} = 3\)[/tex]
- Slope between [tex]\((4, 1)\)[/tex] and [tex]\((1, 1)\)[/tex]: Horizontal line, slope is 0.
This configuration has 0 negative slopes.
### Option 4:
Points: [tex]\[ (1, 4), (2, 1), (5, 1), (4, 4) \][/tex]
- Slope between [tex]\((1, 4)\)[/tex] and [tex]\((2, 1)\)[/tex]: [tex]\(\frac{1 - 4}{2 - 1} = \frac{-3}{1} = -3\)[/tex]
- Slope between [tex]\((2, 1)\)[/tex] and [tex]\((5, 1)\)[/tex]: Horizontal line, slope is 0.
- Slope between [tex]\((5, 1)\)[/tex] and [tex]\((4, 4)\)[/tex]: [tex]\(\frac{4 - 1}{4 - 5} = \frac{3}{-1} = -3\)[/tex]
- Slope between [tex]\((4, 4)\)[/tex] and [tex]\((1, 4)\)[/tex]: Horizontal line, slope is 0.
This configuration has 2 negative slopes.
After evaluating each set, we find that Option 4, which includes the points [tex]\((1, 4), (2, 1), (5, 1), (4, 4)\)[/tex], forms a four-sided figure where two sides have negative slopes.
Thus, the endpoints of the sides of this figure are:
[tex]\[ \boxed{(1,4),(2,1),(5,1),(4,4)} \][/tex]
Let's review each given set of points:
### Option 1:
Points: [tex]\[ (-4, -4), (-4, -1), (-1, -4), (-1, -1) \][/tex]
- Slope between [tex]\((-4, -4)\)[/tex] and [tex]\((-4, -1)\)[/tex]: Vertical line, undefined slope.
- Slope between [tex]\((-4, -1)\)[/tex] and [tex]\((-1, -4)\)[/tex]: [tex]\(\frac{-4 + 1}{-1 + 4} = \frac{-3}{3} = -1\)[/tex]
- Slope between [tex]\((-1, -4)\)[/tex] and [tex]\((-1, -1)\)[/tex]: Vertical line, undefined slope.
- Slope between [tex]\((-1, -1)\)[/tex] and [tex]\((-4, -4)\)[/tex]: [tex]\(\frac{-4 + 1}{-4 + 1} = \frac{-3}{-3} = 1\)[/tex]
This configuration has 1 negative slope.
### Option 2:
Points: [tex]\[ (-2, -4), (-1, -1), (1, -1), (2, -4) \][/tex]
- Slope between [tex]\((-2, -4)\)[/tex] and [tex]\((-1, -1)\)[/tex]: [tex]\(\frac{-1 + 4}{-1 + 2} = \frac{3}{1} = 3\)[/tex]
- Slope between [tex]\((-1, -1)\)[/tex] and [tex]\((1, -1)\)[/tex]: [tex]\(\frac{-1 + 1}{1 + 1} = \frac{0}{2} = 0\)[/tex]
- Slope between [tex]\((1, -1)\)[/tex] and [tex]\((2, -4)\)[/tex]: [tex]\(\frac{-4 + 1}{2 - 1} = \frac{-3}{1} = -3\)[/tex]
- Slope between [tex]\((2, -4)\)[/tex] and [tex]\((-2, -4)\)[/tex]: Horizontal line, slope is 0.
This configuration has 1 negative slope.
### Option 3:
Points: [tex]\[ (1, 1), (2, 4), (5, 4), (4, 1) \][/tex]
- Slope between [tex]\((1, 1)\)[/tex] and [tex]\((2, 4)\)[/tex]: [tex]\(\frac{4 - 1}{2 - 1} = \frac{3}{1} = 3\)[/tex]
- Slope between [tex]\((2, 4)\)[/tex] and [tex]\((5, 4)\)[/tex]: Horizontal line, slope is 0.
- Slope between [tex]\((5, 4)\)[/tex] and [tex]\((4, 1)\)[/tex]: [tex]\(\frac{1 - 4}{4 - 5} = \frac{-3}{-1} = 3\)[/tex]
- Slope between [tex]\((4, 1)\)[/tex] and [tex]\((1, 1)\)[/tex]: Horizontal line, slope is 0.
This configuration has 0 negative slopes.
### Option 4:
Points: [tex]\[ (1, 4), (2, 1), (5, 1), (4, 4) \][/tex]
- Slope between [tex]\((1, 4)\)[/tex] and [tex]\((2, 1)\)[/tex]: [tex]\(\frac{1 - 4}{2 - 1} = \frac{-3}{1} = -3\)[/tex]
- Slope between [tex]\((2, 1)\)[/tex] and [tex]\((5, 1)\)[/tex]: Horizontal line, slope is 0.
- Slope between [tex]\((5, 1)\)[/tex] and [tex]\((4, 4)\)[/tex]: [tex]\(\frac{4 - 1}{4 - 5} = \frac{3}{-1} = -3\)[/tex]
- Slope between [tex]\((4, 4)\)[/tex] and [tex]\((1, 4)\)[/tex]: Horizontal line, slope is 0.
This configuration has 2 negative slopes.
After evaluating each set, we find that Option 4, which includes the points [tex]\((1, 4), (2, 1), (5, 1), (4, 4)\)[/tex], forms a four-sided figure where two sides have negative slopes.
Thus, the endpoints of the sides of this figure are:
[tex]\[ \boxed{(1,4),(2,1),(5,1),(4,4)} \][/tex]
