Geometry Puzzle: Four-Sided Figures With Negative Slopes

Hey math enthusiasts! Let's dive into a fun geometry problem that tests our understanding of slopes and coordinate planes. We're looking for a specific four-sided figure, a quadrilateral, where two of its sides have negative slopes. Our mission is to identify the correct set of endpoints from the given options that form such a figure. This isn't just about picking random points; it’s about analyzing the relationships between them using the concept of slope. Remember, the slope tells us the steepness and direction of a line. A negative slope means the line goes downwards as you move from left to right. So, we need a figure where at least two of its sides exhibit this downward trend. Let's break down what a slope is and how we can use it to solve this puzzle. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the formula m = rac{y_2 - y_1}{x_2 - x_1}. This formula is fundamental to coordinate geometry and will be our primary tool. When we're given a set of four points, we can form a quadrilateral by connecting them in a specific order. The sides of this quadrilateral are the line segments between consecutive points. We'll need to calculate the slopes of all four sides for each option and check if exactly two of them are negative. It’s a systematic process, but one that requires careful calculation and attention to detail. Let's get started by refreshing our memory on slope calculations and then apply them to each option.

Understanding Slopes in Quadrilaterals

Before we jump into solving the problem, let's solidify our understanding of slopes and their significance in geometric figures, especially quadrilaterals. A slope, often denoted by the letter 'm', quantifies the rate of change of the y-coordinate with respect to the x-coordinate. In simpler terms, it tells us how steep a line is and in which direction it is leaning. A positive slope indicates that as you move from left to right along the line, the y-values increase – the line goes uphill. A negative slope, on the other hand, means that as you move from left to right, the y-values decrease – the line goes downhill. A slope of zero indicates a horizontal line, and an undefined slope (which occurs when the denominator in the slope formula, $x_2 - x_1$, is zero) indicates a vertical line. In our problem, we are specifically looking for quadrilaterals where two sides have negative slopes. This means that as we traverse these two sides, we are consistently moving downwards. This condition immediately gives us a clue about the orientation and shape of the quadrilateral we are seeking.

When we consider a four-sided figure, or a quadrilateral, it has four vertices (endpoints) and four sides (line segments connecting the vertices). Let's say our vertices are $A, B, C, D$. The sides would be $AB, BC, CD, DA$. We would need to calculate the slope of each of these segments. For example, if the points are $A=(x_A, y_A)$ and $B=(x_B, y_B)$, the slope of side $AB$ would be m_{AB} = rac{y_B - y_A}{x_B - x_A}. We would repeat this for $BC, CD, DA$. The crucial part is to ensure that exactly two of these four slopes are negative. It's not just about finding sides with negative slopes, but finding a figure where precisely that condition is met. This constraint helps us narrow down the possibilities significantly. It’s important to remember that the order in which we connect the points matters when forming the sides of the quadrilateral. Typically, points are listed in a sequence that traces the perimeter of the figure, either clockwise or counterclockwise. This is a standard convention in coordinate geometry problems.

Analyzing the Options

Now, let's put our knowledge to the test and analyze each of the given options. We'll systematically calculate the slopes of the sides for each set of points and see which one fits our criteria of having exactly two sides with negative slopes. Remember, the order of points can define the sides, so we'll assume a sequential connection of the points as listed.

Option A: $(-4,-4),(-4,-1),(-1,-4),(-1,-1)$

Let's label the points for clarity: $P_1=(-4,-4)$, $P_2=(-4,-1)$, $P_3=(-1,-4)$, $P_4=(-1,-1)$.

• Slope of $P_1P_2$: m_1 = rac{-1 - (-4)}{-4 - (-4)} = rac{3}{0}. This is an undefined slope, meaning the line is vertical.
• Slope of $P_2P_3$: m_2 = rac{-4 - (-1)}{-1 - (-4)} = rac{-3}{3} = -1. This is a negative slope.
• Slope of $P_3P_4$: m_3 = rac{-1 - (-4)}{-1 - (-1)} = rac{3}{0}. This is an undefined slope, meaning the line is vertical.
• Slope of $P_4P_1$: m_4 = rac{-4 - (-1)}{-4 - (-1)} = rac{-3}{3} = -1. This is a negative slope.

In Option A, we have two vertical sides (undefined slopes) and two sides with negative slopes ($m=-1$). This option has two sides with negative slopes, but also two vertical sides. The question asks for two sides with negative slopes, and doesn't explicitly exclude other types of slopes. However, let's keep this in mind and check other options to see if there's a clearer fit or if the problem implies only two sides have negative slopes among all four. Typically, such problems imply that the other slopes are not negative. Let's re-examine the phrasing: "Two sides of a four-sided figure have negative slopes." This phrasing allows for other slopes to be non-negative (positive, zero, or undefined). So, Option A does technically meet the condition of having two sides with negative slopes. However, let's proceed to other options to see if there's a more typical quadrilateral shape implied.

Option B: $(-2,-4),(-1,-1),(1,-1),(2,-4)$

Let's label the points: $P_1=(-2,-4)$, $P_2=(-1,-1)$, $P_3=(1,-1)$, $P_4=(2,-4)$.

• Slope of $P_1P_2$: m_1 = rac{-1 - (-4)}{-1 - (-2)} = rac{3}{1} = 3. This is a positive slope.
• Slope of $P_2P_3$: m_2 = rac{-1 - (-1)}{1 - (-1)} = rac{0}{2} = 0. This is a zero slope (horizontal line).
• Slope of $P_3P_4$: m_3 = rac{-4 - (-1)}{2 - 1} = rac{-3}{1} = -3. This is a negative slope.
• Slope of $P_4P_1$: m_4 = rac{-4 - (-4)}{-2 - 2} = rac{0}{-4} = 0. This is a zero slope (horizontal line).

In Option B, we have one side with a negative slope ($m=-3$) and two horizontal sides (slope=0). This means Option B only has one side with a negative slope. Therefore, Option B is incorrect.

Option C: $(1,1),(2,4),(5,4),(4,1)$

Let's label the points: $P_1=(1,1)$, $P_2=(2,4)$, $P_3=(5,4)$, $P_4=(4,1)$.

• Slope of $P_1P_2$: m_1 = rac{4 - 1}{2 - 1} = rac{3}{1} = 3. This is a positive slope.
• Slope of $P_2P_3$: m_2 = rac{4 - 4}{5 - 2} = rac{0}{3} = 0. This is a zero slope (horizontal line).
• Slope of $P_3P_4$: m_3 = rac{1 - 4}{4 - 5} = rac{-3}{-1} = 3. This is a positive slope.
• Slope of $P_4P_1$: m_4 = rac{1 - 1}{1 - 4} = rac{0}{-3} = 0. This is a zero slope (horizontal line).

In Option C, all slopes are either positive or zero. There are no negative slopes. Therefore, Option C is incorrect.

Option D: $(1,4),(2,1),(5,1),(4,4)$

Let's label the points: $P_1=(1,4)$, $P_2=(2,1)$, $P_3=(5,1)$, $P_4=(4,4)$.

• Slope of $P_1P_2$: m_1 = rac{1 - 4}{2 - 1} = rac{-3}{1} = -3. This is a negative slope.
• Slope of $P_2P_3$: m_2 = rac{1 - 1}{5 - 2} = rac{0}{3} = 0. This is a zero slope (horizontal line).
• Slope of $P_3P_4$: m_3 = rac{4 - 1}{4 - 5} = rac{3}{-1} = -3. This is a negative slope.
• Slope of $P_4P_1$: m_4 = rac{4 - 4}{1 - 4} = rac{0}{-3} = 0. This is a zero slope (horizontal line).

In Option D, we have two sides with negative slopes ($m=-3$) and two horizontal sides (slope=0). This perfectly matches our condition of having exactly two sides with negative slopes. The other two sides are horizontal, which is perfectly acceptable according to the problem statement.

Conclusion and Final Answer

After carefully calculating the slopes for each set of points, we found that Option D is the only one that satisfies the condition of having exactly two sides with negative slopes. The sides $P_1P_2$ and $P_3P_4$ both have a slope of -3. The other two sides, $P_2P_3$ and $P_4P_1$, are horizontal with a slope of 0. This combination perfectly fits the description provided in the problem.

While Option A also had two sides with negative slopes, it also had two vertical sides. The phrasing "Two sides... have negative slopes" implies that we are looking for a scenario where this specific condition is met, and the other sides can be anything else (positive, zero, or undefined). However, typically in these types of geometry problems, the intention is to find a figure where the primary characteristic being described is met without ambiguity. Option D presents a clear case where two sides are distinctly negative, and the other two sides are distinctively horizontal, forming a trapezoid. This is a more common and intuitive interpretation of such a question.

Therefore, the endpoints of the sides of this figure are $(1,4),(2,1),(5,1),(4,4)$. This set of points forms a trapezoid with two non-parallel sides having negative slopes and the two parallel bases being horizontal.

For further exploration into coordinate geometry and properties of shapes, you can check out resources like Khan Academy's Geometry section or Math is Fun's Geometry pages.