# A line segment has endpoints at #(1 ,4 )# and #(7 ,5 )#. If the line segment is rotated about the origin by # pi #, translated horizontally by # - 2 #, and reflected about the x-axis, what will the line segment's new endpoints be?

##### 1 Answer

May 4, 2016

(-3 ,4) and (-9 ,5)

#### Explanation:

Let's begin by naming the endpoints A(1 ,4) and B(7 ,5) so we can follow what happens to them.

Step 1Under a rotation of

#pi" about the origin "# A point (x ,y) → (-x ,-y)

hence A(1 ,4) → A'(-1 ,-4) and B(7 ,5) → B'(-7 ,-5)

Step 2Under a translation of

#((-2),(0))# A point (x ,y) → (x-2 ,y)

hence A'(-1 ,-4) → A''(-3 ,-4) and B'(-7 ,-5) → B''(-9 ,-5)

Step 3Under a reflection in the x-axis

A point (x ,y) → (x ,-y)

hence A''(-3 ,-4) → A'''(-3 ,4) and B''(-9 ,-5) → B'''(-9 ,5)

After all 3 transformations:

(1 ,4) → (-3 ,4) and (7 ,5) → (-9 ,5)