A line segment has endpoints at #(2 , 3)# and #(1 , 2)#. If the line segment is rotated about the origin by #(pi)/2 #, translated vertically by #4#, and reflected about the x-axis, what will the line segment's new endpoints be?

Question

A line segment has endpoints at #(2  ,  3)# and #(1 , 2)#.  If the line segment is rotated about the origin by #(pi)/2   #, translated vertically by #4#, and reflected about the x-axis, what will the line segment's new endpoints be?

Theodore Abad · Accepted Answer

#(-3,-6)" and "(-2,-5)# #"since there are 3 transformations label the endpoints"# #A(2,3)" and "B(1,2)# #color(blue)"first transformation"# #"Under a rotation about the origin of "pi/2# #• " a point "(x,y)to(-y,x)# #rArrA(2,3)toA'(-3,2)# #rArrB(1,2)toB'(-2,1)# #color(blue)"second transformation"# #"under a translation "((0),(4))# #• " a point "(x,y)to(x,y+4)# #rArrA'(-3,2)toA''(-3,6)# #rArrB'(-2,1)toB''(-2,5)# #color(blue)"third transformation"# #"under a reflection in the x-axis"# #• " a point "(x,y)to(x,-y)# #rArrA''(-3,6)toA'''(-3,-6)# #rArrB''(-2,5)toB'''(-2,-5)# #"after all 3 transformations"# #(2,3)to(-3,-6)" and "(1,2)to(-2,-5)#

Aurora Ayala · Answer

undefined To find the new endpoints of the line segment after the described transformations:

1. **Rotation by $ \frac{\pi}{2} $ about the origin**: 
   Applying the rotation matrix for a point $ (x, y) $ by angle $ \theta $ counterclockwise:
   $$ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} $$
   Substituting $ (2, 3) $ and $ (1, 2) $ into this formula, you will get the new coordinates after rotation.

2. **Translation vertically by 4**:
   Adding 4 to the y-coordinates of the points obtained from the rotation.

3. **Reflection about the x-axis**:
   For a point $ (x, y) $, its reflection about the x-axis becomes $ (x, -y) $.

Performing these transformations sequentially will give you the new endpoints of the line segment.