GCSE Maths · Higher · Chapter 7: Transformations, constructions and loci

Congruence, transformations, constructions and loci

Grades 4–9About 85 minutesNo calculator90 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
The point $(3, - 2)$ is reflected in the $x$ -axis. Write down the coordinates of its image.
[1 mark]
Answer

A2
The point $(- 4, 1)$ is translated by the vector $(5 - 3)$ . Write down the coordinates of its image.
[1 mark]
Answer
A3
Write down the equation of the straight line through $(2, 0)$ that is parallel to the $y$ -axis.
[1 mark]
Answer
A4
Name a quadrilateral that has rotational symmetry of order 2 but no lines of symmetry.
[1 mark]
Answer
A5
A triangle has vertices $(1, 1)$ , $(3, 1)$ and $(1, 2)$ . Its image has vertices $(- 1, 1)$ , $(- 3, 1)$ and $(- 1, 2)$ . Describe fully the single transformation.
[2 marks]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

Triangle $A$ has vertices $(1, 1)$ , $(4, 1)$ and $(1, 3)$ . Triangle $B$ has vertices $(- 1, 1)$ , $(- 1, 4)$ and $(- 3, 1)$ . Describe fully the single transformation that maps triangle $A$ onto triangle $B$ .

Hint A rotation needs three things: the angle, the direction and the centre. Use tracing paper to test a centre.

B1
Describe fully the single transformation that maps triangle $P$ onto triangle $Q$ .
[3 marks]
Answer

Example 2

A triangle has vertices $(1, 2)$ , $(2, 2)$ and $(1, 4)$ . It is enlarged by scale factor 3, centre $(0, 1)$ . Write down the coordinates of the vertices of the image.

Hint Find the vector from the centre to each vertex. Multiply it by the scale factor, and add it on to the centre.

B2
The shaded triangle is enlarged by scale factor 2, centre $C (1, 0)$ . Write down the coordinates of the vertices of the image.
[3 marks]
Answer

Example 3

In triangle $A B C$ , $A B = 7$ cm, $B C = 9$ cm and angle $A B C = 5 0^{\circ}$ . In triangle $D E F$ , $D E = 7$ cm, $E F = 9$ cm and angle $D E F = 5 0^{\circ}$ . Are the triangles congruent? Give the reason.

Hint Check where the angle is. Is it between the two given sides?

B3
In triangle $X Y Z$ , $X Y = 8$ cm, angle $X = 4 0^{\circ}$ and angle $Y = 7 5^{\circ}$ . In triangle $P QR$ , $P Q = 8$ cm, angle $P = 4 0^{\circ}$ and angle $Q = 7 5^{\circ}$ . Are the triangles congruent? Give the reason.
[2 marks]
Answer

Example 4

Describe the locus of the points that are 3 cm from a fixed point $P$ .

Hint Mark a few points 3 cm from

P

and look at the shape they make.

B4
A line segment $A B$ is 6 cm long. Describe the locus of the points that are 2 cm from $A B$ .
[2 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Triangle $T$ is shown on the grid.
(a)
Reflect triangle $T$ in the line $y = - x$ . Write down the coordinates of the vertices of the image.
[2 marks]
Answer
(b)
Rotate triangle $T$ through $18 0^{\circ}$ about the point $C (0, 1)$ . Write down the coordinates of the vertices of the image.
[2 marks]
Answer
Total for C1: 4 marks

C2
Describe fully the single transformation that maps triangle $A$ onto triangle $B$ .
[2 marks]
Answer
C3
Shape $S$ is reflected in the $x$ -axis. The image is then reflected in the $y$ -axis.
Describe fully the single transformation that maps shape $S$ onto the final image.
[3 marks]
Answer
C4
The shaded triangle is enlarged by scale factor 3, centre $C (- 4, - 1)$ . Write down the coordinates of the vertices of the image.
[3 marks]
Answer
C5
Describe fully the single transformation that maps triangle $A$ onto triangle $B$ .
[3 marks]
Answer
C6
$A B C D$ is a parallelogram. The diagonal $A C$ is drawn.
Prove that triangle $A B C$ is congruent to triangle $C D A$ .
[3 marks]
C7
The diagram shows a garden $A B C D$ . A fountain is to be put in the garden so that it is less than 3 m from $A$ and nearer to $A B$ than to $A D$ .
Which of the points $P$ , $Q$ , $R$ and $S$ could be the fountain? You must show how you decide.
[3 marks]
Answer
C8
The plan, front elevation and side elevation of a solid are shown.
(a)
Write down the name of the solid.
[1 mark]
Answer
(b)
Work out the volume of the solid.
[2 marks]
Answer
Total for C8: 3 marks
C9
Use a ruler and compasses to construct the perpendicular bisector of $A B$ . You must show all your construction lines.
[2 marks]

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
$A B C$ is an isosceles triangle with $A B = A C$ . $M$ is the midpoint of $B C$ .
Prove that triangle $A B M$ is congruent to triangle $A C M$ , and hence show that $A M$ is perpendicular to $B C$ .
[4 marks]

D2
Triangle $P$ is shown on the grid.
(a)
Reflect triangle $P$ in the $x$ -axis. Label the image $Q$ . Write down the coordinates of the vertices of $Q$ .
[2 marks]
Answer
(b)
Reflect triangle $Q$ in the line $y = x$ . Label the image $R$ . Write down the coordinates of the vertices of $R$ .
[2 marks]
Answer
(c)
Describe fully the single transformation that maps triangle $P$ onto triangle $R$ .
[2 marks]
Answer
Total for D2: 6 marks
D3
Tom says, "If two sides and an angle of one triangle are equal to two sides and an angle of another triangle, then the triangles must be congruent."
Tom is wrong. Explain why.
[2 marks]
D4
Two radio masts $A$ and $B$ are 8 km apart. A signal is picked up at points that are exactly 5 km from $A$ and the same distance from $A$ as from $B$ .
There are two such points. Work out the distance between them.
[3 marks]
Not drawn accurately
Answer
D5
Describe fully the single transformation that maps triangle $A$ onto triangle $B$ .
[3 marks]
Answer
D6
A solid is made from centimetre cubes. It stands on a base 3 cm wide and 2 cm deep. The diagram shows its front elevation and its side elevation.
Work out the largest number of cubes the solid could have.
[3 marks]
Answer
D7
Use a ruler and compasses to construct the bisector of angle $P QR$ . You must show all your construction lines.
[2 marks]
D8
The point $P (5, 2)$ is rotated $9 0^{\circ}$ clockwise about the point $(2, - 1)$ . The image is then translated by the vector $(- 3 4)$ .
Work out the coordinates of the final image of $P$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$
D9
A goat is tied to the corner $G$ of a rectangular shed with a rope 5 m long. The shed is 4 m by 3 m. The goat cannot go into the shed.
Work out the area of ground the goat can reach. Give your answer in terms of $π$ .
[4 marks]
Not drawn accurately
Answer
D10
Describe fully the single transformation that maps triangle $A$ onto triangle $B$ .
[3 marks]
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
A shape is reflected in the line $x = 2$ . The image is then reflected in the line $x = 6$ .
Describe fully the single transformation that has the same effect.
[3 marks]
Hint 1 · What to try
Try it with one point, such as $(0, 0)$ , and then with another point.
Hint 2 · The first line
$(0, 0)$ goes to $(4, 0)$ , then to $(8, 0)$ .
Hint 3 · The full method
Every point moves 8 to the right. The general point $(x, y)$ goes to $(4 - x, y)$ and then to $(12 - (4 - x), y) = (x + 8, y)$ .
Answer

E2
A shape is rotated $9 0^{\circ}$ clockwise about $(0, 0)$ . The image is then rotated $9 0^{\circ}$ clockwise about $(4, 0)$ .
Describe fully the single transformation that has the same effect.
[4 marks]
Hint 1 · What to try
Follow a general point $(x, y)$ through both rotations.
Hint 2 · The first line
About $(0, 0)$ , $(x, y)$ goes to $(y, - x)$ . About $(4, 0)$ , $(a, b)$ goes to $(4 + b, 4 - a)$ .
Hint 3 · The full method
So $(x, y)$ ends at $(4 - x, 4 - y)$ . A point and its image always have midpoint $(2, 2)$ , so this is a half turn about $(2, 2)$ .
Answer
E3
$A B C D$ is a square. $E$ is a point on $B C$ and $F$ is a point on $C D$ , with $B E = C F$ .
Prove that $A E = B F$ and that $A E$ is perpendicular to $B F$ .
[4 marks]
Hint 1 · What to try
Look for two congruent triangles, one with side $A E$ and one with side $B F$ .
Hint 2 · The first line
Triangles $A B E$ and $B C F$ : $A B = B C$ , $B E = C F$ and the angles at $B$ and $C$ are both $9 0^{\circ}$ .
Hint 3 · The full method
So the triangles are congruent (SAS) and $A E = B F$ . Angle $B A E$ = angle $C B F$ . In triangle $A B E$ , angle $B A E$ + angle $A E B = 9 0^{\circ}$ , so angle $C B F$ + angle $A E B = 9 0^{\circ}$ , and the angle where the lines cross is $9 0^{\circ}$ .
E4
An enlargement maps the point $(1, 2)$ to $(7, 8)$ and the point $(2, 0)$ to $(10, 2)$ .
Find the scale factor and the coordinates of the centre of enlargement.
[4 marks]
Hint 1 · What to try
Compare the vector between the two points with the vector between their images.
Hint 2 · The first line
$(1, 2) \to (2, 0)$ is $(1 - 2)$ and $(7, 8) \to (10, 2)$ is $(3 - 6)$ , so the scale factor is 3.
Hint 3 · The full method
If the centre is $(a, b)$ , then $7 = a + 3 (1 - a)$ and $8 = b + 3 (2 - b)$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can prove that two triangles are congruent using SSS, SAS, ASA or RHS.
I can carry out and describe rotations, reflections, translations and enlargements, and combine two of them.
I can construct bisectors, find loci and regions, and use plans and elevations.

Answers and mark scheme

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