Which single transformation that would have the same result as the two transformations (a) rotation by #180^@# about origin and (b) reflection in #y#-axis?

Question

Violet Aldrich · Accepted Answer

The single transformation that would have the same result as the two transformations is the one transforming #(x,y)# to #(x,-y)#, which is nothing but reflection of shape in #x#-axis .
The #x# stays the same, but the #y# changes sign.)

Rotation by #180^@# about the origin, transforms each point #(x,y)# on the shape to the corresponding point #(-x,-y)# (#x and y# both change sign.) When reflected in the #y#-axis, each point #(x,y)# transforms to #(-x,y)# (#x# changes sign, #y# stays the same.) Hence, the single transformation that would have the same result as the two transformations is the one transforming #(x,y)# to #(x,-y)#, #(x,y) rarr (-x,-y) rarr (x,-y)# which is nothing but reflection of shape in the #x#-axis. ( #x# stays the same and #y# changes sign.)

Mason Adams · Answer

#((1,0),(0,-1))# shows a reflection in the #x#-axis.

The #x# -coordinates stay the same, the signs of the #y#-coordinates change.

The two transformations can be described using transformation matrices. The matrix for a rotation of 180° about the origin is #((-1,0),(0,-1))# This has the effect of changing the signs of the #x- and y-# coordinates of all the points. The matrix for a reflection in the #y#-axis is #((-1,0),(0,1))# This has the effect of changing the signs of the #x # -coordinates of all the points, while the #y# -values stay the same. If both transformations take place, the final result is given by: #((-1,0),(0,-1)) xx ((-1,0),(0,1))# #=((1,0),(0,-1))# The effect of this matrix is to keep the #x# -coordinates the same, while changing the signs of the #y# -coordinates - indicating a reflection in the #x#-axis.

Violet Aldrich · Answer

undefined The single transformation that would have the same result as the two transformations (a) rotation by 180 degrees about the origin and (b) reflection in the y-axis is a rotation by 180 degrees about the x-axis.