Consider a triangle that has vertices at A(1,2), B(2,3) and C(4,2). Reflect this triangle in the line through the origin which is inclined at 30 ∘ to the positive x-axis. Find the vertices of the transformed triangle.To solve this question, I rotate this triangle through − 30 ∘ first and then reflect it about x-axis and then rotate through 30 ∘ . I am not sure if this way is correct.

Question

Consider a triangle that has vertices at A(1,2), B(2,3) and C(4,2). Reflect this triangle in the line through the origin which is inclined at       30    ∘   to the positive x-axis. Find the vertices of the transformed triangle.To solve this question, I rotate this triangle through   −      30    ∘   first and then reflect it about x-axis and then rotate through       30    ∘  . I am not sure if this way is correct.

HowOPpodopgtk3 · Accepted Answer

The general approach for finding a 2D matrix for a transform   T  (  p  ) is to see what happens when you transform       e    1    =      [                            1                                      0                      ]   and       e    2    =      [                            0                                      1                      ]   through   T. The transform matrix   M is the 2 by 2 matrix   M  =      [                            T          (                      e            1                    )                          T          (                      e            2                    )                      ]  .So what is   T  (      e    1    ) in your problem? Flipping       [                            1                                      0                      ]   around the       30    o   line gives a unitvector at an angle of       60    o  , or       [                                        1            2                                                                          3                        2                                ]    =  T  (      e    1    )Flipping       [                            0                                      1                      ]   about the        30    o   line gives a unit vector at an angle of   −      30    o  , or       [                                                      3                        2                                                                          −              1                        2                                ]    =  T  (      e    2    ).So altogether your transform is   M  =      [                            T          (                      e            1                    )                          T          (                      e            2                    )                      ]    =      [                                        1            2                                                              3                        2                                                                          3                        2                                                              −              1                        2                                ]  .All that is left is to compute       [                                        1            2                                                              3                        2                                                                          3                        2                                                              −              1                        2                                ]        [                            1                                      2                      ]  ,       [                                        1            2                                                              3                        2                                                                          3                        2                                                              −              1                        2                                ]        [                            2                                      3                      ]  , and       [                                        1            2                                                              3                        2                                                                          3                        2                                                              −              1                        2                                ]        [                            4                                      2                      ]  We skipped the most important step, the first step. The first step was to see if there actually is a matrix that represents your transform   T. The matrix is correct if   T is a linear tranformation, meaning it has 2 properties:-   T  (  P  +  Q  )  =  T  (  P  )  +  T  (  Q  ), the transform of the sum of two vectors is the vector-sum of the transforms-   T  (  c  P  )  =  c  T  (  P  ), scaling a vector by any factor c then transforming it is the same as transforming and then scaling by c.Since we are only reflecting across a line, it&#039;s not too hard to visualize that both of these properties hold, so the matrix   M actually does represent   T.

Consider a triangle that has vertices at A(1,2), B(2,3) and C(4,2). Reflect this triangle in the lin

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