Question

the points ( k(3,2) ), ( l(6,5) ), ( m(3,8) ), and ( n(0,5) ) 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{kl} = )
length of ( overline{kl} = )
slope of ( overline{lm} = )
length of ( overline{lm} = )
slope of ( overline{mn} = )
length of ( overline{mn} = )
slope of ( overline{nk} = )
length of ( overline{nk} = )
quadrilateral ( klmn ) is

Explanation:

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

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_{\overline{KL}}=\frac{5-2}{6-3}=\frac{3}{3}=1$

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

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$d_{\overline{KL}}=\sqrt{(6-3)^2+(5-2)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$

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

$m_{\overline{LM}}=\frac{8-5}{3-6}=\frac{3}{-3}=-1$

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

$d_{\overline{LM}}=\sqrt{(3-6)^2+(8-5)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$

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

$m_{\overline{MN}}=\frac{5-8}{0-3}=\frac{-3}{-3}=1$

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

$d_{\overline{MN}}=\sqrt{(0-3)^2+(5-8)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$

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

$m_{\overline{NK}}=\frac{2-5}{3-0}=\frac{-3}{3}=-1$

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

$d_{\overline{NK}}=\sqrt{(3-0)^2+(2-5)^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$

Step9: Classify the quadrilateral

Opposite slopes are equal (parallel sides), adjacent slopes multiply to $-1$ (perpendicular), all sides equal.

Answer:

slope of $\overline{KL} = 1$
length of $\overline{KL} = 3\sqrt{2}$
slope of $\overline{LM} = -1$
length of $\overline{LM} = 3\sqrt{2}$
slope of $\overline{MN} = 1$
length of $\overline{MN} = 3\sqrt{2}$
slope of $\overline{NK} = -1$
length of $\overline{NK} = 3\sqrt{2}$
Quadrilateral KLMN is a square