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.

Explanation:

Step1: Calculate the length of the first side

The two points \((2,1)\) and \((2, - 4)\) have the same \(x\) - coordinate. Use the distance formula for vertical points \(d=\vert y_2 - y_1\vert\).
\(d_1=\vert1-(-4)\vert=\vert1 + 4\vert = 5\)

Step2: Calculate the length of the second side

The two points \((2,-4)\) and \((-1,-4)\) have the same \(y\) - coordinate. Use the distance formula for horizontal points \(d=\vert x_2 - x_1\vert\).
\(d_2=\vert2-(-1)\vert=\vert2 + 1\vert=3\)

Step3: Calculate the length of the third side

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for the points \((2,1)\) and \((-1,-4)\)
\(x_1 = 2,y_1=1,x_2=-1,y_2=-4\)
\(d_3=\sqrt{(-1 - 2)^2+(-4 - 1)^2}=\sqrt{(-3)^2+(-5)^2}=\sqrt{9 + 25}=\sqrt{34}\)

Step4: Calculate the perimeter

The perimeter \(P\) of a triangle is \(P=d_1 + d_2 + d_3\)
\(P=5 + 3+\sqrt{34}=8+\sqrt{34}\)

Answer:

\(8+\sqrt{34}\)