Question

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

Explanation:

Step1: Calculate the length of \(LN\)

Since \(L(2,4)\) and \(N(-1,4)\) have the same \(y -\)coordinate, use the distance formula \(d=\vert x_2 - x_1\vert\).
\(LN=\vert2-(-1)\vert=\vert2 + 1\vert=3\)

Step2: Calculate the length of \(MN\)

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

Step3: Calculate the length of \(ML\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(M(-2,1)\) and \(L(2,4)\)
\(ML=\sqrt{(2 + 2)^2+(4 - 1)^2}=\sqrt{4^2+3^2}=\sqrt{16 + 9}=5\)

Step4: Calculate the perimeter of \(\triangle LMN\)

Perimeter \(P=LN+MN+ML\)
\(P = 3+\sqrt{10}+5=8+\sqrt{10}\)

Answer:

\(8+\sqrt{10}\text{ units}\) (the fourth option)