Question

Which transformation of AB will produce a line segment A'B' that is parallel to AB?

Answer 1

Options:

Rotation of 180° about point B

Rotation of 90° about point b

Reflection over the x-axis

Translation down 2 units

Answer:

Rotation of 180° about point B

Step-by-step explanation:

Considering coordinates of point B

Assume the coordinates of the line at point B is (x,y)

i.e. $B = (x,y)$

First, we need to determine the slope at point B

Taking coordinates about the origin.

The slope of B is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Where

$(x_1,y_1) = (0,0)$ --- origin

$(x_2,y_2) = (x,y)$

$m = \frac{y_2 - y_1}{x_2 - x_1}$ becomes

$m = \frac{y - 0}{x - 0}$

$m = \frac{y}{x}$

Taking the options one after the other:

Option A.

When rotated by 180°, the resulting coordinates of B would be

$B' = (-x,-y)$

Taking the slope of B'

The slope of B is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Where

$(x_1,y_1) = (0,0)$ --- origin

$(x_2,y_2) = (-x,-y)$

$m = \frac{y_2 - y_1}{x_2 - x_1}$ becomes

$m = \frac{-y - 0}{-x - 0}$

$m = \frac{-y}{-x}$

$m = \frac{y}{x}$

Notice that the slope of B and B' is the same;

$m = \frac{y}{x}$

Hence:

Rotation of 180° about point B  answers the question

There's no need to check for other options