Question

graph and label the quadrilateral with the given vertices. then determine the most precise name for the quadrilateral.

\\(a(-8,-2), b(-4,4), c(4,4), d(8,-2)\\)

choose the correct graph of abcd below.

what is the most precise name for the quadrilateral abcd?

a. the quadrilateral abcd is a trapezoid.
b. the quadrilateral abcd is a rhombus.
c. the quadrilateral abcd is a parallelogram.
d. the quadrilateral abcd is a square.
e. the quadrilateral abcd is a kite.
f. the quadrilateral abcd is an isosceles trapezoid.
g. the quadrilateral abcd is a rectangle.

Answer:

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<reasoning>

Identify the correct graph of ABCD

\[

$$\begin{aligned} &A(-8, -2) \quad (\text{Quadrant III})\\ &B(-4, 4) \quad (\text{Quadrant II})\\ &C(4, 4) \quad (\text{Quadrant I})\\ &D(8, -2) \quad (\text{Quadrant IV}) \end{aligned}$$

\]
Comparing the given coordinates with the options:

• Graph A shows vertices at \(A(-8, -2)\), \(B(-4, 4)\), \(C(4, 4)\), and \(D(8, -2)\).
• Graph B shows a parallelogram.
• Graph C shows a rectangle.
• Graph D shows a trapezoid with different coordinates.

Therefore, Graph A is the correct representation.

Determine the most precise name for ABCD

\[

$$\begin{aligned} &\text{Slope of } BC = \frac{4 - 4}{4 - (-4)} = 0\\ &\text{Slope of } AD = \frac{-2 - (-2)}{8 - (-8)} = 0\\ &\text{Slope of } AB = \frac{4 - (-2)}{-4 - (-8)} = \frac{6}{4} = \frac{3}{2}\\ &\text{Slope of } CD = \frac{-2 - 4}{8 - 4} = \frac{-6}{4} = -\frac{3}{2} \end{aligned}$$

\]
Since \(BC \parallel AD\) and \(AB
ot\parallel CD\), \(ABCD\) is a trapezoid.

To find if it is an isosceles trapezoid, calculate the lengths of the non-parallel sides:
\[

$$\begin{aligned} &AB = \sqrt{(-4 - (-8))^2 + (4 - (-2))^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52}\\ &CD = \sqrt{(8 - 4)^2 + (-2 - 4)^2} = \sqrt{4^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} \end{aligned}$$

\]
Since \(AB = CD\), the trapezoid is an isosceles trapezoid.
</reasoning>

<answer>

Question 1

<mcq-correct>(A) Graph A</mcq-correct>
<mcq-option>(B) Graph B</mcq-option>
<mcq-option>(C) Graph C</mcq-option>
<mcq-option>(D) Graph D</mcq-option>

Question 2

<mcq-option>(A) The quadrilateral ABCD is a trapezoid.</mcq-option>
<mcq-option>(B) The quadrilateral ABCD is a rhombus.</mcq-option>
<mcq-option>(C) The quadrilateral ABCD is a parallelogram.</mcq-option>
<mcq-option>(D) The quadrilateral ABCD is a square.</mcq-option>
<mcq-option>(E) The quadrilateral ABCD is a kite.</mcq-option>
<mcq-correct>(F) The quadrilateral ABCD is an isosceles trapezoid.</mcq-correct>
<mcq-option>(G) The quadrilateral ABCD is a rectangle.</mcq-option>
</answer>

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