Question

the points $b(-6, -6), c(-3, -4), d(-1, 4),$ and $e(-4, 2)$ form a quadrilateral. find the desired slopes and lengths, then fill in the words that best identifies the type of quadrilateral.
answer attempt 1 out of a
slope of $overline{bc}$ = $square$\t\t\t\t\tlength of $overline{bc}$ = $square$
slope of $overline{cd}$ = $square$\t\t\t\t\tlength of $overline{cd}$ = $square$
slope of $overline{de}$ = $square$\t\t\t\t\tlength of $overline{de}$ = $square$
slope of $overline{eb}$ = $square$\t\t\t\t\tlength of $overline{eb}$ = $square$
quadrilateral bcde is

Explanation:

Step1: Calculate slope of $\overline{BC}$

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_{\overline{BC}}=\frac{-4-(-6)}{-3-(-6)}=\frac{2}{3}$

Step2: Calculate length of $\overline{BC}$

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$d_{\overline{BC}}=\sqrt{(-3-(-6))^2+(-4-(-6))^2}=\sqrt{9+4}=\sqrt{13}$

Step3: Calculate slope of $\overline{CD}$

$m_{\overline{CD}}=\frac{4-(-4)}{-1-(-3)}=\frac{8}{2}=4$

Step4: Calculate length of $\overline{CD}$

$d_{\overline{CD}}=\sqrt{(-1-(-3))^2+(4-(-4))^2}=\sqrt{4+64}=\sqrt{68}=2\sqrt{17}$

Step5: Calculate slope of $\overline{DE}$

$m_{\overline{DE}}=\frac{2-4}{-4-(-1)}=\frac{-2}{-3}=\frac{2}{3}$

Step6: Calculate length of $\overline{DE}$

$d_{\overline{DE}}=\sqrt{(-4-(-1))^2+(2-4)^2}=\sqrt{9+4}=\sqrt{13}$

Step7: Calculate slope of $\overline{EB}$

$m_{\overline{EB}}=\frac{-6-2}{-6-(-4)}=\frac{-8}{-2}=4$

Step8: Calculate length of $\overline{EB}$

$d_{\overline{EB}}=\sqrt{(-6-(-4))^2+(-6-2)^2}=\sqrt{4+64}=\sqrt{68}=2\sqrt{17}$

Step9: Classify the quadrilateral

Opposite sides have equal slopes (parallel) and equal lengths, so it is a parallelogram.

Answer:

slope of $\overline{BC}$ = $\frac{2}{3}$
length of $\overline{BC}$ = $\sqrt{13}$
slope of $\overline{CD}$ = $4$
length of $\overline{CD}$ = $2\sqrt{17}$
slope of $\overline{DE}$ = $\frac{2}{3}$
length of $\overline{DE}$ = $\sqrt{13}$
slope of $\overline{EB}$ = $4$
length of $\overline{EB}$ = $2\sqrt{17}$
Quadrilateral BCDE is a parallelogram