Question

use the distance formula to find the lengths of the sides of the quadrilateral. express your answers in simplest radical form or fractions where applicable or appropriate.

statements	reasons
2. kl = units	2. distance formula
lm = units
mn = units
nk = units

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate length of $KL$

For points $K(1,5)$ and $L(6,7)$, $x_1 = 1,y_1 = 5,x_2=6,y_2 = 7$. Then $KL=\sqrt{(6 - 1)^2+(7 - 5)^2}=\sqrt{25 + 4}=\sqrt{29}$ units.

Step3: Calculate length of $LM$

For points $L(6,7)$ and $M(7,3)$, $x_1 = 6,y_1 = 7,x_2=7,y_2 = 3$. Then $LM=\sqrt{(7 - 6)^2+(3 - 7)^2}=\sqrt{1+16}=\sqrt{17}$ units.

Step4: Calculate length of $MN$

For points $M(7,3)$ and $N(3,1)$, $x_1 = 7,y_1 = 3,x_2=3,y_2 = 1$. Then $MN=\sqrt{(3 - 7)^2+(1 - 3)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$ units.

Step5: Calculate length of $NK$

For points $N(3,1)$ and $K(1,5)$, $x_1 = 3,y_1 = 1,x_2=1,y_2 = 5$. Then $NK=\sqrt{(1 - 3)^2+(5 - 1)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}$ units.

Answer:

$KL=\sqrt{29}$ units, $LM=\sqrt{17}$ units, $MN = 2\sqrt{5}$ units, $NK=2\sqrt{5}$ units