Question

determine the most precise name for the quadrilateral $jklm$ with the following vertices: $j(-6,1),k(-2,13),l(10,8),m(6, - 4)$. justify your work by filling in the blanks below: the slope of $overline{jk}=$ blank. the slope of $overline{kl}=$ blank. the slope of $overline{lm}=$ blank. the slope of $overline{mj}=$ blank. from this, i can determine that both pairs of opposite sides are parallel and that only two pairs of consecutive. length of $jk=$ blank. length of $kl=$ blank. length of $lm=$ blank. length of $mj=$ blank. from this, i can determine that both pairs of opposite sides are congruent. therefore, quadrilateral $jklm$ is a parallelogram.

Explanation:

Step1: Calculate slope of $JK$

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $J(-6,1)$ and $K(-2,13)$, $m_{JK}=\frac{13 - 1}{-2-(-6)}=\frac{12}{4} = 3$.

Step2: Calculate slope of $KL$

For points $K(-2,13)$ and $L(10,8)$, $m_{KL}=\frac{8 - 13}{10-(-2)}=\frac{-5}{12}$.

Step3: Calculate slope of $LM$

For points $L(10,8)$ and $M(6,-4)$, $m_{LM}=\frac{-4 - 8}{6 - 10}=\frac{-12}{-4}=3$.

Step4: Calculate slope of $MJ$

For points $M(6,-4)$ and $J(-6,1)$, $m_{MJ}=\frac{1-(-4)}{-6 - 6}=\frac{5}{-12}=-\frac{5}{12}$.

Step5: Analyze parallel - sides

Since $m_{JK}=m_{LM}=3$ and $m_{KL}=m_{MJ}=-\frac{5}{12}$, both pairs of opposite sides are parallel.

Step6: Calculate length of $JK$

The distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $J(-6,1)$ and $K(-2,13)$, $d_{JK}=\sqrt{(-2 + 6)^2+(13 - 1)^2}=\sqrt{16 + 144}=\sqrt{160}=4\sqrt{10}$.

Step7: Calculate length of $KL$

For $K(-2,13)$ and $L(10,8)$, $d_{KL}=\sqrt{(10 + 2)^2+(8 - 13)^2}=\sqrt{144 + 25}=\sqrt{169}=13$.

Step8: Calculate length of $LM$

For $L(10,8)$ and $M(6,-4)$, $d_{LM}=\sqrt{(6 - 10)^2+(-4 - 8)^2}=\sqrt{16+144}=\sqrt{160}=4\sqrt{10}$.

Step9: Calculate length of $MJ$

For $M(6,-4)$ and $J(-6,1)$, $d_{MJ}=\sqrt{(-6 - 6)^2+(1 + 4)^2}=\sqrt{144 + 25}=\sqrt{169}=13$.

Step10: Analyze congruent - sides

Since $d_{JK}=d_{LM}=4\sqrt{10}$ and $d_{KL}=d_{MJ}=13$, both pairs of opposite sides are congruent.

Answer:

Parallelogram