Question

what is the perimeter of △lmn?
○ 8 units
○ 9 units
○ 6 + √10 units
○ 8 + √10 units
(image shows coordinate grid with points n(-1, 4), l(2, 4), m(-2, 1) and triangle lmn)

Explanation:

Step1: Find length of LN

Points L(2,4) and N(-1,4). Since y-coordinates are same, distance is |2 - (-1)| = 3.

Step2: Find length of MN

Points M(-2,1) and N(-1,4). Use distance formula: $\sqrt{(-1 - (-2))^2 + (4 - 1)^2} = \sqrt{1 + 9} = \sqrt{10}$.

Step3: Find length of LM

Points L(2,4) and M(-2,1). Use distance formula: $\sqrt{(2 - (-2))^2 + (4 - 1)^2} = \sqrt{16 + 9} = 5$. Wait, no, wait: Wait, LN is 3 (from x=-1 to x=2, y=4), MN: from (-2,1) to (-1,4): x difference 1, y difference 3, so $\sqrt{1^2 + 3^2}=\sqrt{10}$. LM: from (-2,1) to (2,4): x difference 4, y difference 3? Wait no, L is (2,4), M is (-2,1). So x: 2 - (-2)=4, y: 4 -1=3. So distance is $\sqrt{4^2 + 3^2}=\sqrt{16+9}=5$. Wait, but LN is from (-1,4) to (2,4): x from -1 to 2 is 3 units (since y same). Then MN: (-2,1) to (-1,4): x: -1 - (-2)=1, y:4 -1=3, so distance $\sqrt{1^2 + 3^2}=\sqrt{10}$. LM: (-2,1) to (2,4): x: 2 - (-2)=4, y:4 -1=3, distance $\sqrt{4^2 + 3^2}=5$. Wait, but perimeter is LN + MN + LM = 3 + $\sqrt{10}$ + 5? Wait no, wait LN is 3 (from N(-1,4) to L(2,4): 2 - (-1)=3). MN: from M(-2,1) to N(-1,4): $\sqrt{(-1 - (-2))^2 + (4 - 1)^2}=\sqrt{1 + 9}=\sqrt{10}$. LM: from L(2,4) to M(-2,1): $\sqrt{(2 - (-2))^2 + (4 - 1)^2}=\sqrt{16 + 9}=5$. Wait, but 3 + 5 + $\sqrt{10}$ = 8 + $\sqrt{10}$. Wait, let's recheck LN: N(-1,4), L(2,4). The distance is |2 - (-1)| = 3, correct. MN: M(-2,1), N(-1,4). The horizontal distance is -1 - (-2) = 1, vertical distance is 4 - 1 = 3. So distance is $\sqrt{1^2 + 3^2} = \sqrt{10}$, correct. LM: L(2,4), M(-2,1). Horizontal distance: 2 - (-2) = 4, vertical distance: 4 - 1 = 3. So distance is $\sqrt{4^2 + 3^2} = 5$, correct. So perimeter is 3 + 5 + $\sqrt{10}$ = 8 + $\sqrt{10}$.

Answer:

$8 + \sqrt{10}$ units (the option: 8 + √10 units)