Question

John was visiting four cities that form a rectangle on a coordinate grid at A(0,4), B(4,1), C(3,-1) and D(-1,2). If he visited all the cities in order and ended upwhere he started, what is the distance he traveled? Round your answer to thenearest tenth.

Answer 1

The total distance John travelled can be obtained by finding the sum of the distances between two consequtive points.

The expression for the distance between two points is,

`D=\sqrt[]{(x_2-x_1)^2+(y^2-y_1)^2}`

Here, (x1, y1) and (x2,y2) are the coordinates of two points.

Given, A(0,4), B(4,1), C(3,-1) and D(-1,2) are the coordinates of cities.

The distance beween A(0,4) and B(4,1) is,

`\begin{gathered} AB=\sqrt[]{(4-0)^2+(1-4)^2} \\ =\sqrt[]{16+(-3)^2} \\ =\sqrt[]{16+9} \\ =\sqrt[]{25} \\ =5 \end{gathered}`

The distance beween B(4,1) and C(3,-1) is,

`\begin{gathered} BC=\sqrt[]{(3-4)^2+(-1-1)^2} \\ =\sqrt[]{(-1)^2+(-2)^2} \\ =\sqrt[]{1+4} \\ =\sqrt[]{5} \end{gathered}`

The distance beween C(3,-1) and D(-1,2) is,

`\begin{gathered} CD=\sqrt[]{(-1-3)^2+(2-(-1)})^2 \\ =\sqrt[]{(-4)^2+3^2} \\ =\sqrt[]{16+9} \\ =\sqrt[]{25} \\ =5 \end{gathered}`

The distance beween A(0,4) and D(-1,2) is,

`\begin{gathered} AD=\sqrt[]{(-1-0)^2+(2-4)^2} \\ =\sqrt[]{1+(-2)^2} \\ =\sqrt[]{1+4} \\ =\sqrt[]{5} \end{gathered}`

Now, the total distance John traveled is,

`\begin{gathered} D=AB+BC+CD+AD \\ =5+\sqrt[]{5}+5+\sqrt[]{5} \\ =10+2\sqrt[]{5} \\ =14.5 \end{gathered}`

Therefore, the total distance John traveled rounded to the nearest tenth is 14.5 units.