Question

1. what is the most precise name for quadrilateral (abcd) with vertices (a(-4, 0)), (b(-2, 3)), (c(4, 3)), and (d(2, 0))?

a. parallelogram
b. quadrilateral
c. rhombus
d. rectangle

1. which statement is true?

a. all squares are quadrilaterals.
b. all rectangles are squares.
c. all quadrilaterals are parallelograms.
d. all quadrilaterals are squares.

Explanation:

Question 9

Step1: Calculate side lengths

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

• $AB=\sqrt{(-2+4)^2+(3-1)^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$
• $BC=\sqrt{(4+2)^2+(3-3)^2}=\sqrt{36+0}=6$
• $CD=\sqrt{(2-4)^2+(1-3)^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$
• $DA=\sqrt{(-4-2)^2+(1-1)^2}=\sqrt{36+0}=6$

Step2: Check parallel sides

Calculate slopes: $m=\frac{y_2-y_1}{x_2-x_1}$

• $m_{AB}=\frac{3-1}{-2+4}=1$, $m_{CD}=\frac{1-3}{2-4}=1$ (AB || CD)
• $m_{BC}=\frac{3-3}{4+2}=0$, $m_{DA}=\frac{1-1}{-4-2}=0$ (BC || DA)

Step3: Check right angles

$m_{AB} \times m_{BC}=1 \times 0=0
eq -1$, so no right angles.

• Option A: Squares have 4 sides, so they are quadrilaterals.
• Option B: Rectangles only need 4 right angles; sides don't have to be equal, so not all are squares.
• Option C: Quadrilaterals only need 4 sides; they don't need parallel sides (e.g., trapezoid with one pair, or irregular quadrilaterals), so not all are parallelograms.
• Option D: Most quadrilaterals do not have 4 equal sides and 4 right angles, so this is false.

Answer:

A. parallelogram

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Question 10