Quadrilateral EFGH Analyzing Slopes And Properties

In this comprehensive exploration, we delve into the fascinating world of quadrilaterals, specifically focusing on a quadrilateral named EFGH. This quadrilateral is defined by its vertices: E(-4, 2), F(4, 7), G(8, 1), and H(0, -4). Our primary objective is to meticulously analyze the properties of this quadrilateral, with a particular emphasis on the slopes of its sides. We will investigate the relationships between these slopes to determine whether certain statements about the quadrilateral are true or false. This analysis will not only enhance our understanding of quadrilaterals but also provide valuable insights into the fundamental concepts of coordinate geometry.

Understanding the Basics of Slope

Before we embark on our analysis of quadrilateral EFGH, it is crucial to have a solid grasp of the concept of slope. In coordinate geometry, the slope of a line segment is a numerical value that represents its steepness and direction. It quantifies how much the line rises or falls vertically for every unit of horizontal change. The slope, often denoted by the letter 'm', is calculated using the following formula:

$$m = (y_2 - y_1) / (x_2 - x_1)$$

where (x₁, y₁) and (x₂, y₂) are the coordinates of two distinct points on the line segment. A positive slope indicates an upward inclination, while a negative slope signifies a downward inclination. A slope of zero represents a horizontal line, and an undefined slope corresponds to a vertical line. Furthermore, the slopes of parallel lines are equal, and the slopes of perpendicular lines are negative reciprocals of each other.

Analyzing the Slopes of Sides EH, EF, and GH

Now, let's apply our understanding of slope to quadrilateral EFGH. We will meticulously calculate the slopes of the sides EH, EF, and GH using the slope formula. This will provide us with valuable information about the quadrilateral's shape and properties.

1. Slope of EH:

To determine the slope of EH, we will use the coordinates of points E(-4, 2) and H(0, -4). Substituting these values into the slope formula, we get:

$$m_{EH} = (-4 - 2) / (0 - (-4)) = -6 / 4 = -3/2$$

Therefore, the slope of EH is -3/2. This indicates that EH has a downward inclination, as the slope is negative.

2. Slope of EF:

Next, let's calculate the slope of EF using the coordinates of points E(-4, 2) and F(4, 7). Applying the slope formula, we obtain:

$$m_{EF} = (7 - 2) / (4 - (-4)) = 5 / 8$$

Thus, the slope of EF is 5/8. This positive slope signifies that EF has an upward inclination.

3. Slope of GH:

Now, we will determine the slope of GH using the coordinates of points G(8, 1) and H(0, -4). Plugging these values into the slope formula, we get:

$$m_{GH} = (-4 - 1) / (0 - 8) = -5 / -8 = 5/8$$

Therefore, the slope of GH is 5/8. Similar to EF, GH also has an upward inclination due to its positive slope.

Evaluating the Given Statements

Having calculated the slopes of EH, EF, and GH, we are now equipped to evaluate the given statements about quadrilateral EFGH. Let's examine each statement meticulously and determine its truthfulness based on our calculations.

Statement A: The slope of EH is -8/5.

Our calculations revealed that the slope of EH is -3/2, not -8/5. Therefore, statement A is false. It's crucial to double-check calculations to avoid such errors.

Statement B: The slopes of EF and GH are both 5/8.

We determined that the slope of EF is 5/8 and the slope of GH is also 5/8. This confirms that statement B is true. The equal slopes indicate that EF and GH are parallel lines, a significant property of the quadrilateral.

Now that we have established the slopes of the sides of quadrilateral EFGH, we can delve deeper into its properties and explore its classification. By analyzing the relationships between the slopes and side lengths, we can gain valuable insights into whether the quadrilateral is a parallelogram, rectangle, rhombus, square, or trapezoid. This investigation will further enhance our understanding of the geometric characteristics of EFGH.

Determining Parallelism and Perpendicularity

The slopes of the sides of a quadrilateral provide crucial information about its parallelism and perpendicularity. As we discovered earlier, parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other. By comparing the slopes of the sides of EFGH, we can identify any parallel or perpendicular relationships.

We found that the slopes of EF and GH are both 5/8. This indicates that EF and GH are parallel sides. However, the slope of EH (-3/2) is not equal to the slope of FG (which we haven't calculated yet), suggesting that EH and FG are not parallel. This observation is crucial in classifying the quadrilateral.

To check for perpendicularity, we need to examine whether the product of any two slopes is -1. The product of the slopes of EH (-3/2) and EF (5/8) is -15/16, which is not -1. Similarly, the product of the slopes of EH (-3/2) and GH (5/8) is also -15/16, not -1. This indicates that there are no perpendicular sides in quadrilateral EFGH.

Calculating Side Lengths

In addition to analyzing slopes, calculating the lengths of the sides of the quadrilateral is essential for determining its specific type. We can use the distance formula to find the lengths of EH, EF, GH, and FG. The distance formula is given by:

$$d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)$$

where (x₁, y₁) and (x₂, y₂) are the coordinates of the endpoints of the line segment.

Let's calculate the lengths of the sides of quadrilateral EFGH:

1. Length of EH:

Using the coordinates of E(-4, 2) and H(0, -4), we get:

$$EH = sqrt((0 - (-4))^2 + (-4 - 2)^2) = sqrt(4^2 + (-6)^2) = sqrt(16 + 36) = sqrt(52)$$

2. Length of EF:

Using the coordinates of E(-4, 2) and F(4, 7), we get:

$$EF = sqrt((4 - (-4))^2 + (7 - 2)^2) = sqrt(8^2 + 5^2) = sqrt(64 + 25) = sqrt(89)$$

3. Length of GH:

Using the coordinates of G(8, 1) and H(0, -4), we get:

$$GH = sqrt((0 - 8)^2 + (-4 - 1)^2) = sqrt((-8)^2 + (-5)^2) = sqrt(64 + 25) = sqrt(89)$$

4. Length of FG:

To complete our analysis, we also need to calculate the length of FG using the coordinates of F(4, 7) and G(8, 1):

$$FG = sqrt((8 - 4)^2 + (1 - 7)^2) = sqrt(4^2 + (-6)^2) = sqrt(16 + 36) = sqrt(52)$$

Classifying Quadrilateral EFGH

Based on our calculations, we have the following information:

• Slopes of EF and GH are equal (5/8), indicating that EF and GH are parallel.
• EH and FG are not parallel (slopes are different).
• Side lengths EF and GH are equal (sqrt(89)).
• Side lengths EH and FG are equal (sqrt(52)).

These properties suggest that quadrilateral EFGH is a parallelogram. A parallelogram is a quadrilateral with two pairs of parallel sides. In EFGH, EF and GH are parallel, fulfilling this condition. Additionally, the opposite sides of a parallelogram are equal in length, which is also true for EFGH (EF = GH and EH = FG).

In this comprehensive analysis, we embarked on a journey to unravel the properties of quadrilateral EFGH, defined by its vertices E(-4, 2), F(4, 7), G(8, 1), and H(0, -4). We meticulously calculated the slopes of its sides, determined side lengths, and ultimately classified the quadrilateral as a parallelogram.

Our investigation began with a thorough understanding of the concept of slope, a fundamental tool in coordinate geometry. We then applied the slope formula to calculate the slopes of sides EH, EF, and GH. This analysis revealed that sides EF and GH have equal slopes, indicating their parallelism. We also found that the given statement about the slope of EH being -8/5 is false, emphasizing the importance of accurate calculations.

Furthermore, we delved deeper into the properties of quadrilaterals by calculating side lengths using the distance formula. This allowed us to establish that opposite sides of EFGH are equal in length, a characteristic feature of parallelograms. By combining our findings on slopes and side lengths, we confidently classified quadrilateral EFGH as a parallelogram.

This exploration underscores the power of coordinate geometry in analyzing geometric shapes and understanding their properties. By meticulously applying formulas and concepts, we can gain valuable insights into the relationships between points, lines, and figures in the coordinate plane. The analysis of quadrilateral EFGH serves as a testament to the beauty and applicability of geometric principles in solving real-world problems and furthering our understanding of the world around us.

This detailed analysis not only answers the specific questions about quadrilateral EFGH but also provides a comprehensive understanding of the concepts involved, making it a valuable resource for students and enthusiasts of geometry.