Question

the points u(-7, -7), v(1, -5), w(0, -1), and x(-8, -3) 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 2
slope of \\(\overline{uv}\\) =
length of \\(\overline{uv}\\) =
slope of \\(\overline{vw}\\) =
length of \\(\overline{vw}\\) =
slope of \\(\overline{wx}\\) =
length of \\(\overline{wx}\\) =
slope of \\(\overline{xu}\\) =
length of \\(\overline{xu}\\) =
quadrilateral uvwx is
submit answer

Explanation:

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

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
For $U(-7,-7), V(1,-5)$:
$m_{UV}=\frac{-5-(-7)}{1-(-7)}=\frac{2}{8}=\frac{1}{4}$

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

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$d_{UV}=\sqrt{(1-(-7))^2+(-5-(-7))^2}=\sqrt{8^2+2^2}=\sqrt{64+4}=\sqrt{68}=2\sqrt{17}$

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

For $V(1,-5), W(0,-1)$:
$m_{VW}=\frac{-1-(-5)}{0-1}=\frac{4}{-1}=-4$

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

$d_{VW}=\sqrt{(0-1)^2+(-1-(-5))^2}=\sqrt{(-1)^2+4^2}=\sqrt{1+16}=\sqrt{17}$

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

For $W(0,-1), X(-8,-3)$:
$m_{WX}=\frac{-3-(-1)}{-8-0}=\frac{-2}{-8}=\frac{1}{4}$

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

$d_{WX}=\sqrt{(-8-0)^2+(-3-(-1))^2}=\sqrt{(-8)^2+(-2)^2}=\sqrt{64+4}=\sqrt{68}=2\sqrt{17}$

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

For $X(-8,-3), U(-7,-7)$:
$m_{XU}=\frac{-7-(-3)}{-7-(-8)}=\frac{-4}{1}=-4$

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

$d_{XU}=\sqrt{(-7-(-8))^2+(-7-(-3))^2}=\sqrt{1^2+(-4)^2}=\sqrt{1+16}=\sqrt{17}$

Step9: Classify the quadrilateral

Opposite sides are parallel (equal slopes) and equal in length; adjacent sides have slopes whose product is $\frac{1}{4} \times (-4) = -1$, so they are perpendicular.

Answer:

slope of $\overline{UV}$ = $\frac{1}{4}$
length of $\overline{UV}$ = $2\sqrt{17}$
slope of $\overline{VW}$ = $-4$
length of $\overline{VW}$ = $\sqrt{17}$
slope of $\overline{WX}$ = $\frac{1}{4}$
length of $\overline{WX}$ = $2\sqrt{17}$
slope of $\overline{XU}$ = $-4$
length of $\overline{XU}$ = $\sqrt{17}$
Quadrilateral UVWX is a rectangle