A bicycle repair shop designs a triangular logo, shape \(A\), with vertices \((2,2)\), \((2,5)\) and \((4,2)\). Shape \(A\) is rotated \(90^{\circ}\) anticl...

Shape A has vertices \((2,2)\), \((2,5)\) and \((4,2)\). A rotation of \(90^{\circ}\) anticlockwise about the origin sends \((x,y)\to(-y,x)\); an enlargement of scale factor \(-1\) then sends \((x,y)\to(-x,-y)\). Applying both in sequence, and separately checking that the two steps combine to a single rotation, ties every part of this question together.

(a) Applying \((x,y)\to(-y,x)\) to shape A's vertices: \((2,2)\to(-2,2)\), \((2,5)\to(-5,2)\), \((4,2)\to(-2,4)\). [2 marks]

(b) Applying \((x,y)\to(-x,-y)\) to shape B's vertices: \((-2,2)\to(2,-2)\), \((-5,2)\to(5,-2)\), \((-2,4)\to(2,-4)\). [2 marks]

(c) A \(90^{\circ}\) anticlockwise rotation followed by a \(180^{\circ}\) rotation (the enlargement of scale factor \(-1\)) about the same centre O gives a net turn of \(90^{\circ}+180^{\circ}=270^{\circ}\) anticlockwise, which is the same single transformation as a rotation of \(90^{\circ}\) clockwise about O. [3 marks]

(d) Rotating \((4,2)\) by \(90^{\circ}\) clockwise about O, using \((x,y)\to(y,-x)\), gives \((2,-4)\), which matches the corresponding vertex of shape C found in part (b), confirming the single transformation in part (c). [1 mark]

Verifying a described combined transformation against one actual vertex, as in part (d), is good practice whenever two transformations are combined into one: if the check vertex had not matched, that would flag an error in the description given in part (c).