Question

what is the perimeter of △lmn? n(-1,4) l(2,4) m(-2,1) 6 + √10 units 8 + √10 units 8 units 9 units

Explanation:

Step1: Calculate length of LM

Use distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $L(2,4)$ and $M(-2,1)$, $x_1 = 2,y_1 = 4,x_2=-2,y_2 = 1$. Then $LM=\sqrt{(-2 - 2)^2+(1 - 4)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$.

Step2: Calculate length of MN

For points $M(-2,1)$ and $N(-1,4)$, $x_1=-2,y_1 = 1,x_2=-1,y_2 = 4$. Then $MN=\sqrt{(-1+2)^2+(4 - 1)^2}=\sqrt{1^2+3^2}=\sqrt{1 + 9}=\sqrt{10}$.

Step3: Calculate length of NL

For points $N(-1,4)$ and $L(2,4)$, since $y$-coordinates are the same, $NL=\vert2+1\vert = 3$.

Step4: Calculate perimeter

Perimeter of $\triangle LMN=LM + MN+NL=5+\sqrt{10}+3=8+\sqrt{10}$.

Answer:

$8+\sqrt{10}$ units