Question

find the perimeter of the triangle with these vertices.
$(2,1),(2,-4),(-1,-4)$
give an exact answer (not a decimal approximation).
simplify your answer as much as possible.
scratch area (not part of answer
perimeter:

Explanation:

Step1: Calculate the length of the first side

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for points \((2,1)\) and \((2, - 4)\).
Since \(x_1 = 2,x_2 = 2,y_1 = 1,y_2=-4\), then \(d_1=\sqrt{(2 - 2)^2+(-4 - 1)^2}=\sqrt{0 + 25}=5\)

Step2: Calculate the length of the second side

For points \((2,-4)\) and \((-1,-4)\).
Here \(x_1 = 2,x_2=-1,y_1=-4,y_2 = - 4\), so \(d_2=\sqrt{(-1 - 2)^2+(-4+4)^2}=\sqrt{(-3)^2+0}=3\)

Step3: Calculate the length of the third side

For points \((2,1)\) and \((-1,-4)\).
Using the distance formula \(d_3=\sqrt{(-1 - 2)^2+(-4 - 1)^2}=\sqrt{(-3)^2+(-5)^2}=\sqrt{9 + 25}=\sqrt{34}\)

Step4: Calculate the perimeter

The perimeter \(P=d_1 + d_2 + d_3\), so \(P=5 + 3+\sqrt{34}=8+\sqrt{34}\)

Answer:

\(8+\sqrt{34}\)