Transformations – Year 11 GCSE maths worksheet with answers to download as PDF

Triangle \(T\) has vertices \(A(1,1)\), \(B(4,1)\) and \(C(1,3)\).

1. Reflect \(T\) in the \(y\)-axis. Write the new coordinates. [2 marks]
2. Reflect \(T\) in the line \(y=x\). [2 marks]
3. Reflect \(T\) in the line \(x=5\). [2 marks]
4. Which reflections leave the point \(A(1,1)\) unchanged? [1 mark]

Triangle \(T\) has vertices \(A(1,1)\), \(B(4,1)\) and \(C(1,3)\).

1. Translate \(T\) by the vector \(\begin{pmatrix}-3\\4\end{pmatrix}\). [2 marks]
2. Write the vector that translates the image back to \(T\). [1 mark]
3. Triangle \(T\) is translated so that \(A\) goes to \((6,-2)\). Write the vector and the images of \(B\) and \(C\). [3 marks]

Triangle \(T\) has vertices \(A(1,1)\), \(B(4,1)\) and \(C(1,3)\).

1. Rotate \(T\) by \(90^\circ\) clockwise about the origin. [3 marks]
2. Rotate \(T\) by \(90^\circ\) anticlockwise about the origin. [3 marks]
3. Rotate \(T\) by \(180^\circ\) about the origin. [2 marks]

Point \(P\) is \((4,1)\) and centre \(C\) is \((2,2)\).

1. Write \(P\) relative to \(C\) (the vector \(\overrightarrow{CP}\)). [1 mark]
2. Rotate this vector \(90^\circ\) anticlockwise. [1 mark]
3. Find the image of \(P\) after a \(90^\circ\) anticlockwise rotation about \(C\). [2 marks]
4. Check that the image is the same distance from \(C\) as \(P\). [2 marks]

Triangle \(T\) has vertices \(A(1,1)\), \(B(4,1)\) and \(C(1,3)\).

1. Enlarge \(T\) by scale factor 2, centre the origin. [3 marks]
2. Enlarge \(T\) by scale factor 3, centre \((1,1)\). [3 marks]
3. Find the area of \(T\) and the area of the image in part (b). [3 marks]

Triangle \(T\) has vertices \(A(1,1)\), \(B(4,1)\) and \(C(1,3)\). Describe the single transformation that maps \(T\) onto each image.

1. Image 1: \((-1,1)\), \((-4,1)\), \((-1,3)\). [2 marks]
2. Image 2: \((3,4)\), \((6,4)\), \((3,6)\). [2 marks]
3. Image 3: \((-1,-1)\), \((-4,-1)\), \((-1,-3)\). [3 marks]

A robot arm pivots about the origin. Its tip is at \((6,0)\) and the units are centimetres.

1. Where is the tip after a rotation of \(90^\circ\) anticlockwise? [2 marks]
2. Where is the tip after a rotation of \(180^\circ\)? [1 mark]
3. The arm rotates \(270^\circ\) anticlockwise from the start. Where does it end up? [2 marks]
4. Show that the tip stays on a circle and find the length of the path for one full turn. [3 marks]

An architect draws a rectangle with corners \((2,2)\), \((6,2)\), \((6,5)\) and \((2,5)\) (units in cm). It is enlarged by scale factor 1.5, centre the origin.

1. Find the new coordinates. [3 marks]
2. Find the area of the rectangle before and after. [3 marks]
3. Find the ratio of the two areas. [2 marks]

A logo has a vertex at \(P(3,2)\). It is enlarged with scale factor \(-2\) and centre \((1,1)\).

1. Find the vector from the centre to \(P\). [1 mark]
2. Find the image of \(P\). [3 marks]
3. Describe what a negative scale factor does to the position and size of the shape. [2 marks]
4. The original logo has area 5 cm². Find the image area. [2 marks]

A wheel turns about its centre \(C(3,3)\). A seat is at \(S(6,3)\).

1. Find the position of the seat after a \(90^\circ\) clockwise rotation about \(C\). [3 marks]
2. Find its position after a \(180^\circ\) rotation. [2 marks]
3. The wheel makes one complete turn in 40 seconds. Through what angle does it turn in 15 seconds? [2 marks]
4. Find the speed of the seat in cm/s if the units are centimetres and the radius is 3 cm. [3 marks]

A point \(P(2,2)\) is mapped to \(P'(5,5)\) by an enlargement with scale factor 2.

1. Let the centre be \((a,b)\). Show that \(a+2(2-a)=5\). [2 marks]
2. Find the centre. [2 marks]
3. Check using a second point \(Q(3,2)\) and find its image \(Q'\). [3 marks]

Triangle \(T\) has vertices \((1,1)\), \((4,1)\) and \((1,3)\). It is reflected in the \(y\)-axis and then the image is reflected in the \(x\)-axis.

1. Find the coordinates after both reflections. [3 marks]
2. Describe the single transformation that has the same effect. [2 marks]
3. Does the order of the two reflections matter? Test it. [2 marks]

A tile is a square with corners \((0,0)\), \((2,0)\), \((2,2)\) and \((0,2)\). A floor pattern is made by translating the tile repeatedly.

1. Write the vectors for the tile to the right and the tile above. [2 marks]
2. Write the coordinates of the corners of the tile 3 places to the right and 2 places above. [3 marks]
3. A room is 6 m by 4 m and the tiles are 0.5 m squares. How many tiles are needed? [3 marks]

A pantograph is used to enlarge a drawing by scale factor 2.5. A line on the original is 8 cm long and a shape has area 12 cm².

1. Find the length of the line on the copy. [1 mark]
2. Find the area of the shape on the copy. [2 marks]
3. The copy is a shape of area 75 cm². Find the area of the original. [2 marks]

Point \(P\) is \((2,5)\).

1. Reflect \(P\) in the line \(y=x\) and then reflect the result in the \(y\)-axis. [3 marks]
2. Describe the single rotation that maps \(P\) to the final point. [2 marks]
3. Test your rule on \((3,1)\). [2 marks]

Triangle \(T\) has vertices \((1,1)\), \((4,1)\), \((1,3)\). Triangle \(U\) has vertices \((-1,-1)\), \((-4,-1)\), \((-1,-3)\). Triangle \(V\) has vertices \((1,-1)\), \((4,-1)\), \((1,-3)\).

1. Describe fully the single transformation that maps \(T\) to \(V\). [2 marks]
2. Describe two different transformations that map \(T\) to \(U\). [4 marks]
3. Describe the single transformation that maps \(V\) to \(U\). [2 marks]

Triangle \(PQR\) has \(P(3,2)\), \(Q(5,2)\) and \(R(3,4)\). It is enlarged with scale factor \(-2\) about the centre \((1,1)\).

1. Find the coordinates of \(P'\), \(Q'\) and \(R'\). [6 marks]
2. Find the area of \(PQR\) and of \(P'Q'R'\). [3 marks]
3. Explain why the image is on the opposite side of the centre. [1 mark]

A shape is enlarged by scale factor 3.

1. Its area was 20 cm². Find the new area. [2 marks]
2. A solid of volume 12 cm³ is enlarged by scale factor 3. Find the new volume. [2 marks]
3. A different shape is reduced by scale factor 0.5. A length of 10 cm becomes what, and an area of 40 cm² becomes what? [4 marks]

Triangle \(ABC\) has \(A(2,2)\), \(B(3,2)\), \(C(2,4)\) and its image has \(A'(3,4)\), \(B'(5,4)\), \(C'(3,8)\).

1. Find the scale factor of the enlargement. [2 marks]
2. Find the equations of the lines \(AA'\) and \(BB'\). [4 marks]
3. Find where the lines meet and state the centre. [3 marks]

Triangle \(T\) has vertices \((1,1)\), \((4,1)\), \((1,3)\). It is reflected in the \(x\)-axis, then translated by \(\begin{pmatrix}-2\\5\end{pmatrix}\).

1. Find the final coordinates. [4 marks]
2. Is the final triangle congruent to \(T\)? Explain. [2 marks]
3. Is it possible to describe the combined effect as a single reflection? Test it on point \(A\). [3 marks]

Point \(P\) is \((1,1)\).

1. Reflect \(P\) in the line \(x=3\) and then reflect the image in the line \(x=7\). [3 marks]
2. Describe the single transformation that has the same effect and relate it to the distance between the mirrors. [3 marks]
3. Reflect \((2,0)\) in the line \(y=0\) and then in the line \(y=x\). Describe the single transformation, relating the angle to the angle between the mirrors. [4 marks]