Question

Line AB with endpoints A(4,4) and B(5,-1) is reflected in the line y=x to create its image Line A’B’. Graph A’B’. Then, find the perimeter of the figure formed by the segments Line AB , Line B’B and Line AB’ to the nearest tenth. The grey line was already graphed for me already. I started by graphing the points 4,4 and 5,-1. Then I reflected it in the line y=x but I’m having trouble figuring out the perimeter.

Answer 1

• Given the line AB whose endpoints are:

$\begin{gathered} A\mleft(4,4\mright) \\ B\mleft(5,-1\mright) \end{gathered}$

You know that the line of reflection is:

$y=x$

By definition, the rule for that Reflection is:

$(x,y)\rightarrow\mleft(y,x\mright)$

Then, the endpoints of the Image A'B' are:

$\begin{gathered} A(4,4)\rightarrow A^{\prime}(4,4) \\ B(5,-1)\rightarrow B^{\prime}(-1,5) \end{gathered}$

Therefore, you can notice that the lines of AB and A'B' are graphed correctly.

• According to the information given in the exercise, you must find the perimeter of the figure formed by these segments:

$\begin{gathered} AB \\ B^{\prime}B \\ AB^{\prime} \end{gathered}$

Look at the following picture, where you can see the segments that form a triangle:

You can see in the picture a triangle that is formed by the segments AB, B’B and AB'.

To find the length of each segment by using the formula for calculating the distance between two points:

$d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}$

Where the points are:

$(x_1,y_1);(x_2,y_2)_{}$

Therefore, knowing the endpoints of each segment of the triangle, you get:

- Length of the side AB:

$AB=\sqrt[]{(5-4)^2+(-1-4)^2}$$\begin{gathered} AB=\sqrt[]{(1)^2+(-5)^2} \\ AB=\sqrt[]{26} \end{gathered}$

- Length of the side AB':

$A^{}B^{\prime}=\sqrt[]{(-1-4)^2+(5-4)^2}=\sqrt[]{26}$

- Length of the side B'B:

$B^{\prime}B=\sqrt[]{(-1-5)^2+(5-(-1))^2}=6\sqrt[]{2}$

Knowing the length of each side, you can add them in order to find the perimeter:

$P=\sqrt[]{26}+\sqrt[]{26}+6\sqrt[]{2}\approx18.7\text{ }units$

Therefore, the answer is:

$P\approx18.7\text{ }units$