(a) A triangle has vertices at the points A(3,−1), B(−2,1) and C(2,3). Suppose that the triangle is to be moved so that B is at the origin and BA lies along the positive x-axis. One isometry that achieves this transformation is the composite of a translation followed by a rotation. (You may find it helpful to sketch the triangle.)

(i) Determine the translation that moves B to the origin, giving your answer in the form . Write down a formal definition of this translation in two-line notation.

(ii) Find the images A' of A and C' of C under the translation in part (i)

(iii) Let rθ be the rotation that completes the required transformation, where θ lies in the interval (−π,π]. Find the exact values of tan θ,cos θand sin θ, and hence write down a formal definition of rθ using two-line notation. (There is no need to work out the value of the angle θ.)

(iv) Find the coordinates of the images of A' and C' under the rotation rθ. Give your answers as exact values.

I have done (i) and (ii) see below, but am unsure about how to work out the rotation without knowing what angle it rotates by. Thanks for your pointers.

(i)

(ii) A' (5,-2) and C' (4,2)

(iii) ?