Apply these transformations to points in the plane and polygons.

Triangle \(ABC\) has vertices \(A(2, 1)\), \(B(5, 1)\) and \(C(3, 4)\). The triangle is transformed by the matrix \(M = \begin{pmatrix} 1 & 2 \\ -1 & 3 \end{pmatrix}\) to form triangle \(A'B'C'\). Determine the coordinates of the vertices of the transformed triangle \(A'B'C'\). Show evidence of the matrix multiplication in your solution.

A triangle with vertices \(A(2, 1)\), \(B(4, 1)\) and \(C(3, 3)\) is transformed by the matrix \(M = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\). Determine the coordinates of the image of vertex \(C\) under this transformation.

A triangle in the complex plane has vertices at \(z_1 = 2 + i\), \(z_2 = 4 + 3i\), and \(z_3 = 1 + 4i\). The triangle is transformed by the mapping \(w = iz + 3\). What are the coordinates of the image of vertex \(z_2\) under this transformation?

A quadrilateral PQRS has vertices P(2, 1), Q(5, 2), R(6, 5), and S(3, 4). The quadrilateral undergoes three successive transformations: • First, a reflection in the line y = x • Second, a rotation of 90° anticlockwise about the origin • Third, a dilation of factor 2 from the origin (a) Determine the single transformation matrix that represents the first two transformations combined. (2 marks) (b) Determine the coordinates of Q'', the image of vertex Q after the first two transformations. (1 mark) (c) Hence determine the coordinates of Q''' and R''', the final images of vertices Q and R after all three transformations. (2 marks)

A quadrilateral ABCD has vertices A(2, 1), B(5, 1), C(5, 4) and D(2, 4). The quadrilateral undergoes three successive transformations: • First, a rotation of π/4 radians anticlockwise about the origin. • Second, a reflection in the line y = -x. • Third, a dilation of factor √2 from the origin. Determine the coordinates of the final image of vertex C after all three transformations have been applied.