<em>Options:</em>
<em>Rotation of 180° about point B
</em>
<em>Rotation of 90° about point b
</em>
<em>Reflection over the x-axis
</em>
<em>Translation down 2 units</em>
<em></em>
Answer:
Rotation of 180° about point B
<em></em>
Step-by-step explanation:
Considering coordinates of point B
Assume the coordinates of the line at point B is (x,y)
i.e. ![B = (x,y)](https://tex.z-dn.net/?f=B%20%3D%20%28x%2Cy%29)
First, we need to determine the slope at point B
Taking coordinates about the origin.
The slope of B is:
![m = \frac{y_2 - y_1}{x_2 - x_1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By_2%20-%20y_1%7D%7Bx_2%20-%20x_1%7D)
Where
--- origin
![(x_2,y_2) = (x,y)](https://tex.z-dn.net/?f=%28x_2%2Cy_2%29%20%3D%20%28x%2Cy%29)
becomes
![m = \frac{y - 0}{x - 0}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By%20-%200%7D%7Bx%20-%200%7D)
![m = \frac{y}{x}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By%7D%7Bx%7D)
Taking the options one after the other:
Option A.
When rotated by 180°, the resulting coordinates of B would be
![B' = (-x,-y)](https://tex.z-dn.net/?f=B%27%20%3D%20%28-x%2C-y%29)
Taking the slope of B'
The slope of B is:
![m = \frac{y_2 - y_1}{x_2 - x_1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By_2%20-%20y_1%7D%7Bx_2%20-%20x_1%7D)
Where
--- origin
![(x_2,y_2) = (-x,-y)](https://tex.z-dn.net/?f=%28x_2%2Cy_2%29%20%3D%20%28-x%2C-y%29)
becomes
![m = \frac{-y - 0}{-x - 0}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B-y%20-%200%7D%7B-x%20-%200%7D)
![m = \frac{-y}{-x}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B-y%7D%7B-x%7D)
![m = \frac{y}{x}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By%7D%7Bx%7D)
Notice that the slope of B and B' is the same;
![m = \frac{y}{x}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7By%7D%7Bx%7D)
<em>Hence:</em>
<em>Rotation of 180° about point B answers the question</em>
<em></em>
<em>There's no need to check for other options</em>