Exercise solution 4, page 522 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) A rotation is a transformation that turns a shape around a fixed point, called the centre of rotation. The shape keeps exactly the same size and shape, so rotation is an isometry. Only its position and orientation change.

B) To rotate a shape on a coordinate grid, it is helpful to work with its vertices, that is, the corner points of the shape. Read each vertex carefully from the grid, then rotate each one around the given centre. After that, join the new points in the same order.

C) A rotation of \(90^\circ\) anti-clockwise means a quarter-turn to the left. Anti-clockwise is the opposite direction to the hands of a clock. It is important not to confuse this with a \(90^\circ\) clockwise rotation, which would turn the shape to the right.

D) When the centre of rotation is not the origin, the easiest method is to look at each point relative to the centre. In other words, find how far the point is across and how far it is up from the centre.

If a point is moved using a \(90^\circ\) anti-clockwise rotation about a centre, then a horizontal and vertical movement \((a,b)\) becomes \((-b,a)\).

For example, suppose the centre is \((4,2)\) and a vertex is at \((6,5)\). From the centre, the point is:

\[ \begin{aligned} (6-4,\;5-2) &= (2,3) \end{aligned} \]

After a \(90^\circ\) anti-clockwise rotation, \((2,3)\) becomes \((-3,2)\). So the new point is:

\[ \begin{aligned} (4,2)+(-3,2) &= (1,4) \end{aligned} \]

E) Another good way to think about rotation is with left and up movements from the centre. For a \(90^\circ\) anti-clockwise turn:

right becomes up,
up becomes left,
left becomes down,
down becomes right.

For instance, if a point is 3 squares to the right and 1 square up from the centre, after a \(90^\circ\) anti-clockwise rotation it will be 1 square left and 3 squares up from the centre.

F) Distances from the centre stay the same during a rotation. This means each rotated vertex must be the same distance from the centre as it was before. This is a useful check: if a point was 2 squares away horizontally and 4 vertically from the centre before rotating, the new point should still have those same sizes of movement, but in a turned position.

G) Once all the new vertices have been found, plot them carefully and join them in order. Make sure the rotated shape has the same side lengths as the original. A common mistake is to rotate only one point correctly but then draw the rest of the shape by guessing.

H) Always check three things at the end: the direction of the turn (anti-clockwise), the angle of the turn (\(90^\circ\)), and the centre of rotation. If one of these is wrong, the final shape will be in the wrong place even if the drawing looks neat.