Question

2 ?
e (-2, 3) f (1, 6)
d = \sqrt{(1 + 2)^2 + (6 - 3)^2}
d = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}
\overline{fg}
f (1, 6) g (4, 3)
d = \sqrt{(4 - 1)^2 + (3 - 6)^2}
d = \sqrt{3^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2}
g (4, 3) h (1, 0)
d = \sqrt{(1 - 4)^2 + (0 - 3)^2}
d = \sqrt{(-3)^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2}
\overline{eh}
e (-2, 3) h (1, 0)
d = \sqrt{(1 + 2)^2 + (0 - 3)^2}
d = \sqrt{3^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2}
3 \overline{ef}\parallel\overline{gh}
\overline{ef}
e (-2, 3) f (1, 6)
m = \frac{6 - 3}{1 + 2} = \frac{3}{3} = 1
\overline{gh}
g (4, 3) h (1, 0)
m = \frac{0 - 3}{1 - 4} = \frac{-3}{-3} = 1
4 ?
\overline{eh}
e(-2, 3) h (1, 0)
m = \frac{0 - 3}{1 + 2} = \frac{-3}{3} = -1
\overline{fg}
f (1, 6) g (4, 3)
m = \frac{3 - 6}{4 - 1} = \frac{-3}{3} = -1
5 \overline{ef} and \overline{gh} are perpendicular to \overline{fg}
5 the slope of \overline{ef} and \overline{gh} is 1. the slope of \overline{fg} is -1.
6 ?
6 the slope of \overline{fg} and \overline{eh} is -1. the slope of \overline{gh} is 1
7 quadrilateral efgh is a square

1. all sides are congruent, opposite sides are parallel, and adjacent sides are perpendicular

which of the following completes statement 4 of the proof?
\overline{ef}, \overline{fg}, \overline{gh}, and \overline{eh} are congruent
\overline{ef}\parallel\overline{gh}
\overline{eh}\parallel\overline{fg}
\overline{fg} and \overline{eh} are perpendicular to \overline{gh}

Explanation:

Step1: Analyze the slopes

For two lines to be parallel, their slopes must be equal. The slope of \(\overline{EH}\) is calculated as \(m_{EH}=\frac{0 - 3}{1+2}=\frac{-3}{3}=-1\), and the slope of \(\overline{FG}\) is \(m_{FG}=\frac{3 - 6}{4 - 1}=\frac{-3}{3}=-1\).

Step2: Determine the relationship

Since \(m_{EH}=m_{FG}=-1\), by the definition of parallel lines (lines with equal slopes are parallel), \(\overline{EH}\parallel\overline{FG}\).

Answer:

\(\overline{EH}\parallel\overline{FG}\)