A triangle ABC has vertices A(1, 2), B(3, 4), and C(5, 6). Apply a dilation with a scale factor of 2 centered at the origin. Determine the new coordinates for A.

An equilateral triangle XYZ has vertices X(0, 0), Y(4, 0), and Z(2, 4√3). Perform a translation with a vector [3, -2]. What are the new coordinates for X?

Given a point P(4, 6), apply a dilation with a scale factor of 0.5 centered at the origin. Determine the new coordinates of point P'.

Consider a square with vertices A(0, 0), B(0, 2), C(2, 2), and D(2, 0). Translate the square by -3 units in the x-direction and +1 unit in the y-direction using a matrix. Find the new coordinates for B.