Question

name: _______________________ date: __________ homework 5.5 a) graph and label quadrilateral bugs with vertices b(- 4, - 2), u(0, 6), g(6, 4) and s(8, - 6) b) prove that quadrilateral bugs is a trapezoid.

Explanation:

Part (a)

Step 1: Plot Point B

Locate \( x = -4 \) and \( y = -2 \) on the coordinate grid. Mark this point as B.

Step 2: Plot Point U

Locate \( x = 0 \) and \( y = 6 \) on the coordinate grid. Mark this point as U.

Step 3: Plot Point G

Locate \( x = 6 \) and \( y = 4 \) on the coordinate grid. Mark this point as G.

Step 4: Plot Point S

Locate \( x = 8 \) and \( y = -6 \) on the coordinate grid. Mark this point as S.

Step 5: Connect the Points

Draw line segments to connect B to U, U to G, G to S, and S to B to form quadrilateral BUGS. Label each vertex.

Part (b)

Step 1: Recall the Trapezoid Definition

A trapezoid is a quadrilateral with at least one pair of parallel sides. Parallel sides have equal slopes.

Step 2: Calculate the Slope of BU

The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For points B\((-4, -2)\) and U\((0, 6)\):
\( m_{BU} = \frac{6 - (-2)}{0 - (-4)} = \frac{8}{4} = 2 \)

Step 3: Calculate the Slope of UG

For points U\((0, 6)\) and G\((6, 4)\):
\( m_{UG} = \frac{4 - 6}{6 - 0} = \frac{-2}{6} = -\frac{1}{3} \)

Step 4: Calculate the Slope of GS

For points G\((6, 4)\) and S\((8, -6)\):
\( m_{GS} = \frac{-6 - 4}{8 - 6} = \frac{-10}{2} = -5 \)

Step 5: Calculate the Slope of SB

For points S\((8, -6)\) and B\((-4, -2)\):
\( m_{SB} = \frac{-2 - (-6)}{-4 - 8} = \frac{4}{-12} = -\frac{1}{3} \)

Step 6: Identify Parallel Sides

Compare the slopes: \( m_{UG} = -\frac{1}{3} \) and \( m_{SB} = -\frac{1}{3} \). Since these slopes are equal, sides UG and SB are parallel. Thus, BUGS has at least one pair of parallel sides, so it is a trapezoid.

Answer:

(for part b):
Quadrilateral BUGS is a trapezoid because sides UG and SB have equal slopes (\( -\frac{1}{3} \)), meaning they are parallel, satisfying the trapezoid definition.