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 \(N(-1,4)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(y_1 = y_2 = 4\), so \(LN=\vert2-(-1)\vert=3\)

Step2: Calculate the length of \(MN\)

For \(M(-2,1)\) and \(N(-1,4)\), \(d=\sqrt{(-1+2)^2+(4 - 1)^2}=\sqrt{1 + 9}=\sqrt{10}\)

Step3: Calculate the length of \(ML\)

For \(M(-2,1)\) and \(L(2,4)\), \(d=\sqrt{(2 + 2)^2+(4 - 1)^2}=\sqrt{16+9}=5\)

Step4: Calculate the perimeter

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

Answer:

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