A line segment has endpoints at #(1 ,6 )# and #(5 ,1 )#. If the line segment is rotated about the origin by # pi #, translated horizontally by # - 4 #, and reflected about the y-axis, what will the line segment's new endpoints be?

The starting points' coordinates are:

and

I step - rotate about origin.

and

II step - translation

and

and

Finally we can say that the new coordinates are:

To find the new endpoints of the line segment after the given transformations:

1. Rotation by ( \pi ) about the origin: The rotation matrix for a rotation by ( \theta ) about the origin is: [ \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} ]

For ( \theta = \pi ): [ \begin{bmatrix} \cos(\pi) & -\sin(\pi) \ \sin(\pi) & \cos(\pi) \end{bmatrix} = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} ]

2. Translation horizontally by -4: Given points: ( (x_1, y_1) = (1, 6) ) and ( (x_2, y_2) = (5, 1) ) After translation: [ x_1' = x_1 - 4 = 1 - 4 = -3 ] [ x_2' = x_2 - 4 = 5 - 4 = 1 ]

3. Reflection about the y-axis: After reflection: [ x_1'' = -x_1' = 3 ] [ x_2'' = -x_2' = -1 ]

Now, using the rotation matrix: [ \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 3 \ 6 \end{bmatrix} = \begin{bmatrix} -3 \ -6 \end{bmatrix} ] [ \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \times \begin{bmatrix} -1 \ 1 \end{bmatrix} = \begin{bmatrix} 1 \ -1 \end{bmatrix} ]

So, the new endpoints after all transformations are: Endpoint 1: ( (3, -6) ) Endpoint 2: ( (1, -1) )