A quadrilateral has vertices $E(-4,2), F(4,7), G(8,1)$, and $H(0,-4)$. Which statements are true? Check all that apply.
A. The slope of $\overline{EH}$ is $-\frac{8}{5}$.
B. The slopes of $\overline{EF}$ and $\overline{GH}$ are both $\frac{5}{8}$.
C. $\overline{FG}$ is perpendicular to $\overline{GH}$.
D. Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.
E. Quadrilateral EFGH is a rectangle because all angles are right angles.
Answer (2)
The true statements are that the slopes of EF and G H are both 8 5 , and quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel. All other statements are false.
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Calculate the slopes of all sides of the quadrilateral using the formula m = x 2 − x 1 y 2 − y 1 .
Compare the calculated slopes to the statements provided to determine if they are true or false.
Check if opposite sides have equal slopes to verify if the quadrilateral is a parallelogram.
Check if adjacent sides have slopes that are negative reciprocals of each other to verify if the quadrilateral is a rectangle.
Identify the true statements based on the calculations and comparisons.
The true statements are: The slopes of EF and G H are both 8 5 , and Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.
Explanation
Problem Analysis We are given a quadrilateral EFGH with vertices E(-4, 2), F(4, 7), G(8, 1), and H(0, -4). We need to determine which of the given statements are true.
Calculate Slopes First, let's calculate the slope of each side of the quadrilateral using the formula: m = x 2 − x 1 y 2 − y 1
Slope of E H : m E H = 0 − ( − 4 ) − 4 − 2 = 4 − 6 = − 2 3 = − 1.5
Slope of EF :
m EF = 4 − ( − 4 ) 7 − 2 = 8 5 = 0.625
Slope of G H :
m G H = 8 − 0 1 − ( − 4 ) = 8 5 = 0.625
Slope of FG :
m FG = 8 − 4 1 − 7 = 4 − 6 = − 2 3 = − 1.5
Evaluate Statements Now, let's evaluate the given statements:
Statement 1: The slope of E H is − 5 8 .
Our calculation shows m E H = − 2 3 . Therefore, this statement is false .
Statement 2: The slopes of EF and G H are both 8 5 .
Our calculations show m EF = 8 5 and m G H = 8 5 . Therefore, this statement is true .
Statement 3: FG is perpendicular to G H .
For two lines to be perpendicular, the product of their slopes must be -1. Let's check: m FG ⋅ m G H = − 2 3 ⋅ 8 5 = − 16 15
Since − 16 15 = − 1 , FG is not perpendicular to G H . Therefore, this statement is false .
Statement 4: Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.
For EFGH to be a parallelogram, EF ∥ G H and E H ∥ FG . This means their slopes must be equal.
We have m EF = m G H = 8 5 and m E H = m FG = − 2 3 . Therefore, this statement is true .
Statement 5: Quadrilateral EFGH is a rectangle because all angles are right angles.
For EFGH to be a rectangle, adjacent sides must be perpendicular. We already checked that FG is not perpendicular to G H . Also, EF is not perpendicular to E H because m EF ⋅ m E H = 8 5 ⋅ − 2 3 = − 16 15 = − 1 . Therefore, this statement is false .
Conclusion Based on our calculations and analysis, the true statements are:
The slopes of EF and G H are both 8 5 .
Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.
Examples
Understanding slopes and geometric properties like parallelism and perpendicularity is crucial in various real-world applications. For example, architects use these concepts to design buildings with parallel walls and perpendicular intersections, ensuring structural stability and aesthetic appeal. Similarly, city planners use these principles to design road layouts, ensuring efficient traffic flow and safe intersections. In computer graphics, these concepts are used to create realistic 3D models and animations.
