A line segment has endpoints at #(9 ,6 )# and #(5 ,3)#. If the line segment is rotated about the origin by #pi #, translated vertically by #2 #, and reflected about the x-axis, what will the line segment's new endpoints be?

Question

A line segment has endpoints at #(9 ,6    )# and #(5  ,3)#.  If the line segment is rotated about the origin by #pi     #, translated vertically by #2  #, and reflected about the x-axis, what will the line segment's new endpoints be?

Theodore Abad · Accepted Answer

#(9,6)to(-9,4),(5,3)to(-5,1)#

Since there are 3 transformations to be performed, name the endpoints A(9 ,6) and B(5 ,3). This will enable us to 'track' the position of the points after each transformation.

First transformation Under a rotation about the origin of #pi#

a point (x ,y) → (-x ,-y)

hence A(9 ,6) → A'(-9 ,-6) and B(5 ,3) → B'(-5 ,-3)

Second transformation Under a translation #((0),(2))#

a point (x ,y) → (x ,y+2)

hence A'(-9 ,-6) → A''(-9 ,-4) and B'(-5 ,-3) → B''(-5 ,-1)

Third transformation Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

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

Thus after the 3 transformations.

#(9,6)to(-9,4)" and " (5,3)to(-5,1)#

Evelyn Arboleda · Answer

undefined The new endpoints of the line segment after rotation by $ \pi $, translation vertically by 2 units, and reflection about the x-axis are:

Endpoint 1: $ (9 \cos(\pi) - 6 \sin(\pi), 6 \cos(\pi) + 9 \sin(\pi) + 2) $

Endpoint 2: $ (5 \cos(\pi) - 3 \sin(\pi), 3 \cos(\pi) + 5 \sin(\pi) + 2) $

Simplified, these become:

Endpoint 1: $ (-9, 9 + 2) $

Endpoint 2: $ (-5, 5 + 2) $

So, the new endpoints are:

Endpoint 1: $ (-9, 11) $

Endpoint 2: $ (-5, 7) $