Question

use the image to answer the question
use the coordinates to compute the perimeter of the triangle

Explanation:

Step1: Calculate the length of the side between \((1,1)\) and \((5,1)\)

Since the \(y -\)coordinates are the same (\(y = 1\)), use the distance formula \(d=\vert x_2 - x_1\vert\). Here \(x_1 = 1,x_2=5\), so \(d_1=\vert5 - 1\vert=4\)

Step2: Calculate the length of the side between \((1,1)\) and \((3,4)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 1,y_1 = 1,x_2=3,y_2 = 4\), so \(d_2=\sqrt{(3 - 1)^2+(4 - 1)^2}=\sqrt{4 + 9}=\sqrt{13}\)

Step3: Calculate the length of the side between \((3,4)\) and \((5,1)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 3,y_1 = 4,x_2=5,y_2 = 1\), so \(d_3=\sqrt{(5 - 3)^2+(1 - 4)^2}=\sqrt{4+9}=\sqrt{13}\)

Step4: Calculate the perimeter

The perimeter \(P=d_1 + d_2 + d_3\). Substitute \(d_1 = 4,d_2=\sqrt{13},d_3=\sqrt{13}\), so \(P=4 + 2\sqrt{13}\approx4+2\times3.606=4 + 7.212 = 11.212\)

Answer:

\(4 + 2\sqrt{13}\approx11.21\)