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 (1)
Calculate the slope of EH and determine that the statement 'The slope of E H is − 5 8 ' is false.
Calculate the slopes of EF and GH and confirm that the statement 'The slopes of EF and G H are both 8 5 ' is true.
Calculate the slope of FG and check if FG is perpendicular to G H , determining that the statement is false.
Verify that EFGH is a parallelogram because both pairs of opposite sides are parallel, making the statement true.
Check if EFGH is a rectangle by checking if adjacent sides are perpendicular, determining that the statement is false.
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 Let's analyze the given quadrilateral EFG H 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 by calculating slopes and checking for parallelism and perpendicularity.
Calculate Slope of EH First, we calculate the slope of E H using the formula m = x 2 − x 1 y 2 − y 1 .
m E H = 0 − ( − 4 ) − 4 − 2 = 4 − 6 = − 2 3 = − 1.5 The statement 'The slope of E H is − 5 8 ' is false since − 2 3 = − 5 8 .
Calculate Slopes of EF and GH Next, we calculate the slope of EF :
m EF = 4 − ( − 4 ) 7 − 2 = 8 5 And the slope of G H :
m G H = 0 − 8 − 4 − 1 = − 8 − 5 = 8 5 The statement 'The slopes of EF and G H are both 8 5 ' is true.
Calculate Slope of FG and Check Perpendicularity Now, we calculate the slope of FG :
m FG = 8 − 4 1 − 7 = 4 − 6 = − 2 3 To check if FG is perpendicular to G H , we multiply their slopes: m FG ⋅ m G H = − 2 3 ⋅ 8 5 = − 16 15 Since − 16 15 = − 1 , the statement ' FG is perpendicular to G H ' is false.
Check if EFGH is a Parallelogram To determine if EFG H is a parallelogram, we need to check if opposite sides are parallel. We already know that m EF = m G H = 8 5 , so EF ∥ G H .
Now we need to check if E H ∥ FG . We have m E H = − 2 3 and m FG = − 2 3 , so E H ∥ FG .
Since both pairs of opposite sides are parallel, EFG H is a parallelogram. Thus, the statement 'Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel' is true.
Check if EFGH is a Rectangle To determine if EFG H is a rectangle, we need to check if adjacent sides are perpendicular. We can check if EF is perpendicular to FG :
m EF ⋅ m FG = 8 5 ⋅ ( − 2 3 ) = − 16 15 Since − 16 15 = − 1 , EF is not perpendicular to FG . Therefore, EFG H is not a rectangle. The statement 'Quadrilateral EFGH is a rectangle because all angles are right angles' is false.
Final Answer Based on our calculations, 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
In architecture, determining if a quadrilateral is a parallelogram or rectangle is crucial for ensuring structural stability and aesthetic appeal. For instance, when designing a building's facade, architects use these geometric principles to ensure that opposite sides are parallel (parallelogram) or that all angles are right angles (rectangle), which affects the building's overall look and structural integrity. By calculating slopes and checking for parallelism and perpendicularity, architects can accurately plan and execute designs that are both visually pleasing and structurally sound. This ensures that the building meets safety standards and maintains its intended appearance over time.
