Question

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

Explanation:

Step1: Calculate length of MN

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $M(-2,1)$ and $N(-1,4)$, we have $x_1=-2,y_1 = 1,x_2=-1,y_2 = 4$. Then $MN=\sqrt{(-1+2)^2+(4 - 1)^2}=\sqrt{1 + 9}=\sqrt{10}$.

Step2: Calculate length of NL

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

Step3: Calculate length of LM

For points $L(2,4)$ and $M(-2,1)$, we can use the distance formula. First, the horizontal distance is $\vert2 + 2\vert=4$ and the vertical distance is $\vert4 - 1\vert = 3$. Then $LM=\sqrt{(2 + 2)^2+(4 - 1)^2}=\sqrt{16+9}=5$.

Step4: Calculate the perimeter

The perimeter $P$ of $\triangle LMN$ is $P=MN+NL+LM=\sqrt{10}+3 + 5=8+\sqrt{10}$.

Answer:

$8+\sqrt{10}$ units