Year 8 transformations worksheet answers to download as PDF

These Year 8 transformations worksheet answers give a full worked solution to all 20 questions of the transformations worksheet. Try each question yourself first, then check your method step by step.

← Back to the worksheet

Answer to Question 1
1. \((x,y)\to(x,-y)\): \((1,-1)\), \((3,-1)\), \((1,-4)\)
2. \((x,y)\to(-x,y)\): \((-1,1)\), \((-3,1)\), \((-1,4)\)
3. \((x,y)\to(y,x)\): \((1,1)\), \((1,3)\), \((4,1)\)
4. \(A\) is 2 to the left of the line \(x=3\), so the image is 2 to the right: \((5,1)\)
Each reflection keeps the triangle the same size and shape; only its position and orientation change.
Answer to Question 2
A translation by \(\begin{pmatrix}a\\b\end{pmatrix}\) adds a to every x-coordinate and b to every y-coordinate: positive a moves right, positive b moves up.
1. Add 4 to \(x\) and 2 to \(y\): \((5,3)\), \((7,3)\), \((5,6)\)
2. \((-2,2)\), \((0,2)\), \((-2,5)\)
3. \((1,-4)\), \((3,-4)\), \((1,-1)\)
4. \(-1-2=-3\) and \(7-3=4\): \(\begin{pmatrix}-3\\4\end{pmatrix}\)
Answer to Question 3
Rotations about the origin follow simple rules: 90° anticlockwise \((x,y)\to(-y,x)\), 180° \((x,y)\to(-x,-y)\), 90° clockwise \((x,y)\to(y,-x)\).
1. \((x,y)\to(-y,x)\): \((-1,1)\), \((-1,3)\), \((-4,1)\)
2. \((x,y)\to(-x,-y)\): \((-1,-1)\), \((-3,-1)\), \((-1,-4)\)
3. \((x,y)\to(y,-x)\): \((1,-1)\), \((1,-3)\), \((4,-1)\)
4. Rotation \(90^\circ\) anticlockwise about the origin (\((3,1)\to(-1,3)\)).
A full description of a rotation needs three things: the angle, the direction and the centre.
Answer to Question 4
When the centre of enlargement is the origin, multiply every coordinate by the scale factor.
1. Multiply each coordinate by 2: \((2,2)\), \((6,2)\), \((2,8)\)
2. \((3,3)\), \((9,3)\), \((3,12)\)
3. Halve each coordinate: \((1,1)\), \((3,1)\), \((1,4)\)
4. Base \(=2\times2=\) 4 units.
A scale factor between 0 and 1 makes the shape smaller, which is still called an enlargement in maths.
Answer to Question 5
Multiply each length by the scale factor. Perimeters scale by the same factor as lengths, but areas scale by the square of the scale factor.
1. \(12\) cm by \(9\) cm.
2. Original \(14\) cm; image \(42\) cm; ratio \(1:3\) (same as the scale factor).
3. Original \(12\) cm²; image \(108\) cm²; ratio \(1:9\) (the scale factor squared).
4. \(10\times4^2=\) 160 cm²
Check (c): \(12\times9=108\), and \(108\div12=9=3^{2}\).
Answer to Question 6
1. Every point moves 5 to the left: translation by \(\begin{pmatrix}-5\\0\end{pmatrix}\).
2. \(y\) changes sign: reflection in the \(x\)-axis.
3. Each coordinate doubles: enlargement, scale factor 2, centre \((0,0)\).
4. (a) and (b) are congruent (same size). (c) is similar but not congruent.
Reflections, rotations and translations give congruent images; an enlargement with a scale factor other than 1 gives a similar image.
Answer to Question 7
1. \(\begin{pmatrix}0\\1\end{pmatrix}\)
2. Column 5 is 4 steps right and row 3 is 2 steps up: \(\begin{pmatrix}4\\2\end{pmatrix}\)
3. Width: \(300\div20=15\) tiles. Height: \(240\div20=12\) tiles. Total 180 tiles.
4. Single tiles: \(180\times1.85=£333\). Boxes: 18 boxes \(\times16=£288\). Boxes are cheaper by £45.
Always check that the number of tiles is a whole number in each direction: here 20 cm divides exactly into both 300 cm and 240 cm.
Answer to Question 8
1. Reflecting in the y-axis changes the sign of x: \((1,1)\), \((4,3)\), \((5,0)\), \((3,-2)\).
2. The y-axis is the line where x is always 0: \(x=0\).
3. Lengths are multiplied by the scale factor: \(10\times2.5=\)25 cm.
4. Areas are multiplied by the scale factor squared: \(24\times2.5^{2}=24\times6.25=\)150 cm².
Answer to Question 9
1. Using \((x,y)\to(-y,x)\): \((0,60)\to\)\((-60,0)\), a quarter turn from the top to the left.
2. Half a turn: \((0,60)\to\)\((0,-60)\), at the bottom of the wheel.
3. Three quarters of a turn: \((0,60)\to\)\((60,0)\), on the right.
4. 90° is a quarter of a full turn, so it takes \(30\div4=\)7.5 minutes.
Answer to Question 10
1. \(21\times1.41=\) 29.6 cm and \(29.7\times1.41=\) 41.9 cm (A3 is 29.7 by 42).
2. \(50\times1.41^2=50\times1.988=\) 99.4 cm²
3. \(1.41\times0.71=1.0011\approx1\), so the page returns to almost exactly its original size.
4. \(1.41^2=1.988\approx2\): yes, the area is almost doubled.
The scale factor 1.41 is close to \(\sqrt{2}\), which is why A3 has twice the area of A4.
Answer to Question 11
1. 19 m = 1 900 cm. Divide by 76: \(1900\div76=\)25.0 cm.
2. Multiply by 76: \(25\times76=1900\) cm, which is 19.0 m.
3. Areas scale by \(76^{2}=5776\). 0.9 m² = 9 000 cm², so \(9000\div5776\approx\)1.6 cm².
4. \(40\times76=3040\) cm, which is 30.4 m.
Answer to Question 12
1. The lamp post’s shadow is 4 times the child’s: scale factor 4.
2. Heights scale in the same way: \(1.5\times4=\)6 m.
3. The tree’s shadow is 6.4 times the child’s, so its height is \(1.5\times6.4=\)9.6 m.
4. The scale factor between child and lamp post is still 4: \(2\times4=\)8 m.
Answer to Question 13
1. \((7,4)\), \((9,4)\), \((9,5)\), \((7,5)\)
2. \((4,2)\to(9,5)\) is a move of \(\begin{pmatrix}5\\3\end{pmatrix}\): yes, the same.
3. Relative to \((2,1)\) the corner is \((2,0)\); clockwise \(90^\circ\) gives \((0,-2)\), so the image is \((2,-1)\).
4. 2 m² both before and after: rotation keeps area unchanged.
Answer to Question 14
1. \(8\times2=\) 16 m and \(5\times2=\) 10 m.
2. \(16\times10=\) 160 m²
3. 1 cm² on the plan is \(2\times2=4\) m², so \(3\times4=\) 12 m²
4. Twice as big means 1 cm to 1 m, which is 1 : 100.
Areas on the plan scale by the square of the length factor: 1 cm² stands for \(2\times2=4\) m².
Answer to Question 15
1. \(T_1\): \((-1,1)\), \((-3,1)\), \((-1,4)\)
2. \(T_2\): \((-1,-4)\), \((-3,-4)\), \((-1,-1)\)
3. No simple translation works, since the triangle is flipped (mirror image). The overall move is a glide reflection: reflect in the \(y\)-axis and translate down 5.
4. Yes: reflections and translations preserve length and angles, so \(T_2\) is congruent to \(T\).
Answer to Question 16
1. \(A\) is 1 right and 1 up from the centre. Image: 3 right, 3 up: \((4,4)\)
2. \(B\): \((1+3\times3,1+3\times1)=\) \((10,4)\). \(C\): \((1+3\times1,1+3\times2)=\) \((4,7)\).
3. Object base \(AB=2\), image base \(=6\); object height 1, image height 3. All lengths are 3 times bigger.
4. Object: \(\frac12\times2\times1=1\). Image: \(\frac12\times6\times3=9\). Area is 9 times bigger.
Answer to Question 17
1. Sides: 2, 4, 6, 8. Areas: 4, 16, 36, 64.
2. Area \(=(2k)^2=4k^2\).
3. \(4k^2=144\), \(k^2=36\), \(k=6\).
4. Area at \(k=4\) is 64 and at \(k=2\) is 16: the ratio is 4, which is \((4\div2)^2\).
Doubling the scale factor always multiplies the area by \(2^{2}=4\).
Answer to Question 18
1. \((1,0)\), \((2,-3)\), \((0,-5)\), \((-1,-3)\)
2. Final image: \((-1,0)\), \((-2,-3)\), \((0,-5)\), \((1,-3)\). Single transformation: rotation \(180^\circ\) about the origin.
3. The original is not symmetrical about the \(y\)-axis since \((1,0)\) has no mirror point \((-1,0)\).
4. \(y=0\) (the \(x\)-axis).
Reflecting in both axes, one after the other, always has the same effect as a half-turn about the origin: \((x,y)\to(-x,-y)\).
Answer to Question 19
1. Ali is correct: \((x,y)\to(-x,y)\) gives \((-1,1)\), \((-2,1)\), \((-1,3)\) exactly.
2. Bea: \((1,1)\to(-1,1)\) works but \((2,1)\to(0,1)\), which is wrong. Cal: \((1,1)\to(-1,-1)\), wrong. Dev: \((1,1)\to(1,-1)\), wrong.
3. Bea’s translation sends \((2,1)\) to \((0,1)\), but the correct image is \((-2,1)\). A translation cannot flip a shape over, so it cannot match a reflection.
Answer to Question 20
1. Multiply each coordinate by \(-2\): \((-2,-2)\), \((-6,-2)\), \((-2,-4)\)
2. The image is on the opposite side of the centre and is upside down (rotated \(180^\circ\)).
3. Original area \(=\frac12\times2\times1=1\). Image area \(=1\times(-2)^2=\) 4.
4. A rotation of \(180^\circ\) about the centre.
Points on the image are 2 times as far from the centre as the object, but on the opposite side of it.

How to use these Year 8 transformations worksheet answers

First, try every question on the worksheet without looking. Then compare your working with the solution line by line, not only the final answer. When your answer is different, find the first step that changes, because that is where the mistake is. After that, read the matching part of the transformations lesson again. Finally, check that the method is secure with the quiz or the test.

Also, the Year 8 transformations worksheet answers are available as a PDF, so you can mark printed work quickly. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, these Year 8 transformations worksheet answers show every step, so nothing is left unexplained.