Exercise solution 2, page 525 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) A translation moves a shape without turning it, flipping it or changing its size. Every point moves by the same vector.

If the vector is written as \(\begin{pmatrix} a \\ b \end{pmatrix}\), then:

\[ \text{move } a \text{ units horizontally and } b \text{ units vertically} \]

A positive horizontal value means move right, and a negative horizontal value means move left. A positive vertical value means move up, and a negative vertical value means move down.

For example, a translation by \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\) means 4 units right and 3 units down.

B) When translating points on a coordinate grid, add the vector to the coordinates of each point.

If a point \((x,y)\) is translated by \(\begin{pmatrix} a \\ b \end{pmatrix}\), its image is:

\[ (x+a,\; y+b) \]

For example, if \((2,5)\) is translated by \(\begin{pmatrix} -1 \\ -4 \end{pmatrix}\), then:

\[\begin{aligned} (2,5) &\to (2-1,\;5-4) \\ &= (1,1) \end{aligned}\]

This is useful for checking where key vertices of the shape go.

C) A rotation of \(180^\circ\) turns a shape half a turn about a given centre. After a \(180^\circ\) rotation, each point ends up directly opposite the centre, at the same distance from it.

This means the centre of rotation is the midpoint of a point and its image.

If a point \(P\) rotates \(180^\circ\) about a centre \(C\), then \(C\) is halfway between \(P\) and \(P'\).

D) On a grid, a \(180^\circ\) rotation about a centre \((h,k)\) can be found using coordinates.

If \((x,y)\) is rotated \(180^\circ\) about \((h,k)\), the image is:

\[ (2h-x,\; 2k-y) \]

For example, rotating \((6,1)\) by \(180^\circ\) about \((2,-3)\):

\[\begin{aligned} x' &= 2\times 2 - 6 = -2 \\ y' &= 2\times (-3) - 1 = -7 \end{aligned}\]

So the image is \((-2,-7)\).

E) A single transformation means replacing two steps by one equivalent transformation. In this type of question, a translation followed by a \(180^\circ\) rotation often combines into another \(180^\circ\) rotation, but with a different centre.

To describe a transformation fully, you must give the correct type and the necessary information:

translation: the vector

rotation: the angle, direction if needed, and centre

reflection: the mirror line

enlargement: the scale factor and centre

F) A good method is to track one or two vertices carefully.

For each vertex:

first apply the translation, then apply the rotation to the new point.

After that, compare the starting shape and the final shape to decide the single transformation.

Be careful to keep the transformations in the correct order, because doing them in a different order can give a different result.

G) When reading the diagram, make sure you use the axes correctly:

\(x\)-coordinates tell you horizontal position,

\(y\)-coordinates tell you vertical position.

A point further left has a smaller \(x\)-coordinate, and a point lower down has a smaller \(y\)-coordinate.

Counting squares accurately is essential, especially when the vector contains negative numbers.