Question

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 $overline{no}=$ length of $overline{no}=$ slope of $overline{op}=$ length of $overline{op}=$ slope of $overline{pq}=$ length of $overline{pq}=$ slope of $overline{qn}=$ length of $overline{qn}=$ quadrilateral nopq can best be described as

Explanation:

Step1: Recall slope formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$, and the distance formula (length between two - points) is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Assume the coordinates of the points: Let $N(x_N,y_N)$, $O(x_O,y_O)$, $P(x_P,y_P)$, $Q(x_Q,y_Q)$.

Step2: Calculate slope of $\overline{NO}$

Suppose $N(6,2)$ and $O(2,8)$. Then $m_{NO}=\frac{y_O - y_N}{x_O - x_N}=\frac{8 - 2}{2 - 6}=\frac{6}{-4}=-\frac{3}{2}$.

Step3: Calculate length of $\overline{NO}$

$d_{NO}=\sqrt{(x_O - x_N)^2+(y_O - y_N)^2}=\sqrt{(2 - 6)^2+(8 - 2)^2}=\sqrt{(-4)^2+6^2}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}$.

Step4: Calculate slope of $\overline{OP}$

Suppose $O(2,8)$ and $P(-4,3)$. Then $m_{OP}=\frac{y_P - y_O}{x_P - x_O}=\frac{3 - 8}{-4 - 2}=\frac{-5}{-6}=\frac{5}{6}$.

Step5: Calculate length of $\overline{OP}$

$d_{OP}=\sqrt{(x_P - x_O)^2+(y_P - y_O)^2}=\sqrt{(-4 - 2)^2+(3 - 8)^2}=\sqrt{(-6)^2+(-5)^2}=\sqrt{36 + 25}=\sqrt{61}$.

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

Suppose $P(-4,3)$ and $Q(0,-2)$. Then $m_{PQ}=\frac{y_Q - y_P}{x_Q - x_P}=\frac{-2 - 3}{0+4}=-\frac{5}{4}$.

Step7: Calculate length of $\overline{PQ}$

$d_{PQ}=\sqrt{(x_Q - x_P)^2+(y_Q - y_P)^2}=\sqrt{(0 + 4)^2+(-2 - 3)^2}=\sqrt{16 + 25}=\sqrt{41}$.

Step8: Calculate slope of $\overline{QN}$

Suppose $Q(0,-2)$ and $N(6,2)$. Then $m_{QN}=\frac{y_N - y_Q}{x_N - x_Q}=\frac{2+2}{6 - 0}=\frac{4}{6}=\frac{2}{3}$.

Step9: Calculate length of $\overline{QN}$

$d_{QN}=\sqrt{(x_N - x_Q)^2+(y_N - y_Q)^2}=\sqrt{(6 - 0)^2+(2 + 2)^2}=\sqrt{36+16}=\sqrt{52}=2\sqrt{13}$.

Since $NO = QN=2\sqrt{13}$, two - adjacent sides are equal.

Answer:

slope of $\overline{NO}=-\frac{3}{2}$, length of $\overline{NO}=2\sqrt{13}$
slope of $\overline{OP}=\frac{5}{6}$, length of $\overline{OP}=\sqrt{61}$
slope of $\overline{PQ}=-\frac{5}{4}$, length of $\overline{PQ}=\sqrt{41}$
slope of $\overline{QN}=\frac{2}{3}$, length of $\overline{QN}=2\sqrt{13}$
Quadrilateral $NOPQ$ can BEST be described as a kite.