A triangle has vertices at the points

A(−6,9),B(6,3) andC(9,−6).

Suppose that the triangle is to be moved so that

Bis at the origin

and

BClies along the positivex-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

Bto the origin, giving your

answer in the form

ta,b. Write down a formal definition of this

translation in two-line notation.

2

(ii) Find the coordinates of the images

A!ofAandC!ofCunder the

translation in part (a)(i).

(iii) Let

rθbe the rotation that completes the required

transformation, where

θlies in the interval (−π, π]. Find the

exact values of tan

θ, cosθand sinθ, and hence write down a

formal definition of

rθusing 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!andC!under the

rotation

rθ. Give your answers as exact values.

(v) Write down a formal definition of the composite transformation,

that is, the result of the translation in part (a)(i) followed by the

rotation in part (a)(iii).

I've been stuck on this all day! I have t-6,3 for part (i) and found A to be (-12,6) and C to be (3,-9) for part (ii).

Other than that I'm totally lost!

Any help would be really appreciated, thanks.