Question

what is the perimeter of $\triangle lmn$? \bigcirc 8 units \bigcirc 9 units \bigcirc 6 + \sqrt{10} units \bigcirc 8 + \sqrt{10} units

Explanation:

Step1: Calculate length of LN

Points $L(2,4)$ and $N(-1,4)$ have the same y-coordinate. Subtract x-values:
$LN = |2 - (-1)| = 3$

Step2: Calculate length of MN

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ for $M(-2,1)$ and $N(-1,4)$:
$MN = \sqrt{(-1 - (-2))^2 + (4 - 1)^2} = \sqrt{(1)^2 + (3)^2} = \sqrt{1+9} = \sqrt{10}$

Step3: Calculate length of LM

Use distance formula for $L(2,4)$ and $M(-2,1)$:
$LM = \sqrt{(2 - (-2))^2 + (4 - 1)^2} = \sqrt{(4)^2 + (3)^2} = \sqrt{16+9} = \sqrt{25} = 5$

Step4: Sum lengths for perimeter

Add the three side lengths:
$\text{Perimeter} = LN + MN + LM = 3 + \sqrt{10} + 5$

Answer:

$8 + \sqrt{10}$ units