Final answer:
Option 4 is the correct set of endpoints for a four-sided figure with two sides having negative slopes. The slopes of the lines formed between these endpoints can be calculated, and two of them are negative, satisfying the condition of the question.
Step-by-step explanation:
In this question, we have to determine which set of endpoints form sides with negative slopes for a four-sided figure. A negative slope means that the line moves down the y-axis as the x-value increases. To find which option gives two sides with negative slopes, we need to calculate the slope between each pair of endpoints. The slope is given by the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line
- Option 1: Slopes cannot be calculated because the points do not form a four-sided figure without intersecting lines.
- Option 2: Also not a valid option since two points are the same, indicating no line segment.
- Option 3: This forms a four-sided figure but there are no sides with negative slopes.
- Option 4: Using the slope formula, the side between (1,4) and (2,1) has a slope of (1 - 4) / (2 - 1) = -3 and the side between (2,1) and (5,1) has a zero slope. Then moving from the side between (5,1) and (4,4), we get a positive slope of (4 - 1) / (4 - 5) = 3 / -1 = -3. Thus, option 4 has two sides with negative slopes.
The correct answer is Option 4, which has the endpoints (1,4), (2,1), (5,1), and (4,4) forming two sides of the figure with negative slopes.