Question

quadrilateral tuvw has vertices at t(3, -5), u(5, 5), v(-5, 7), and w(-7, -3). is tuvw a square? justify your answer. yes, all sides are congruent, and \\(\overline{uv}\\) and \\(\overline{vw}\\) are perpendicular. no, \\(\overline{tu}\\) and \\(\overline{uv}\\) are not congruent. no, \\(\overline{tw}\\) and \\(\overline{vw}\\) are not congruent. no, \\(\overline{tu}\\) and \\(\overline{tw}\\) are not perpendicular.

Explanation:

Step1: Calculate length of $\overline{TU}$

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$\overline{TU}=\sqrt{(5-3)^2+(5-(-5))^2}=\sqrt{2^2+10^2}=\sqrt{4+100}=\sqrt{104}$

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

$\overline{UV}=\sqrt{(-5-5)^2+(7-5)^2}=\sqrt{(-10)^2+2^2}=\sqrt{100+4}=\sqrt{104}$

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

$\overline{VW}=\sqrt{(-7-(-5))^2+(-3-7)^2}=\sqrt{(-2)^2+(-10)^2}=\sqrt{4+100}=\sqrt{104}$

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

$\overline{WT}=\sqrt{(3-(-7))^2+(-5-(-3))^2}=\sqrt{10^2+(-2)^2}=\sqrt{100+4}=\sqrt{104}$

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

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_{TU}=\frac{5-(-5)}{5-3}=\frac{10}{2}=5$

Step6: Calculate slope of $\overline{UV}$

$m_{UV}=\frac{7-5}{-5-5}=\frac{2}{-10}=-\frac{1}{5}$

Step7: Check perpendicularity of $\overline{TU}$ & $\overline{UV}$

Perpendicular slopes multiply to -1: $5\times(-\frac{1}{5})=-1$, so they are perpendicular.

Step8: Check slope of $\overline{VW}$

$m_{VW}=\frac{-3-7}{-7-(-5)}=\frac{-10}{-2}=5$
It is perpendicular to $\overline{UV}$ ($-\frac{1}{5}\times5=-1$), confirming adjacent sides are perpendicular.

Answer:

Yes, all sides are congruent, and $\overline{UV}$ and $\overline{VW}$ are perpendicular.