Triangle ABC has vertices A(1, 1), B(4, 1) and C(1, 3). Triangle ABC is translated by the vector (3, -2) to give triangle A'B'C'.
(a)Write down the coordinates of A'.(1)
(b)Write down the coordinates of B'.(1)
(c)Write down the coordinates of C'.(1)
2
Point P has coordinates (5, 2).
(a)Write down the coordinates of the image of P after a reflection in the x-axis.(1)
(b)Write down the coordinates of the image of P after a reflection in the y-axis.(1)
3
State the order of rotational symmetry of each shape.
(a)A square.(1)
(b)A regular pentagon.(1)
(c)An isosceles triangle that is not equilateral.(1)
4
Point A has coordinates (2, 5) and point B has coordinates (-3, 4). All rotations are about the origin O.
(a)A is rotated 90 degrees clockwise about O. Write down the coordinates of the image of A.(1)
(b)B is rotated 180 degrees about O. Write down the coordinates of the image of B.(1)
5
Shape P has vertices (1, 1), (3, 1) and (1, 2). Shape Q has vertices (4, 3), (6, 3) and (4, 4). Describe fully the single transformation that maps shape P onto shape Q.
(2)
6
Triangle ABC has vertices A(1, 2), B(4, 2) and C(1, 5).
(a)Reflect triangle ABC in the line y = x. Write down the coordinates of A'.(1)
(b)Write down the coordinates of B'.(1)
(c)Write down the coordinates of C'.(1)
7
Rectangle ABCD has vertices A(1, 1), B(3, 1), C(3, 2) and D(1, 2). Enlarge rectangle ABCD by scale factor 2, centre (0, 0). Write down the coordinates of A', B', C' and D'.
(2)
8
Triangle ABC has vertices A(2, 2), B(4, 2) and C(2, 4). Triangle ABC is enlarged by scale factor 3, centre (1, 1). Find the coordinates of A', B' and C'.
(3)
9
Shape R has vertices (2, 1), (2, 4) and (4, 1). Shape S has vertices (-1, 2), (-4, 2) and (-1, 4). Describe fully the single transformation that maps shape R onto shape S.
(3)
10
Point A has coordinates (2, 3). A is translated by the vector (1, -4) to give point A1. A1 is then reflected in the y-axis to give point A2. Find the coordinates of A2.
(2)
11
On a coordinate grid where distances are measured in metres, a drone starts at point (2, 3). It moves according to the vector (6, 8), then according to the vector (-2, 5).
Diagram NOT accurately drawn
(a)Work out the coordinates of the drone's final position.(2)
(b)Calculate the direct straight-line distance between the drone's start and finish positions, giving your answer in metres to 1 decimal place.(3)
12
Rectangle ABCD has vertices A(2, 4), B(6, 4), C(6, 8) and D(2, 8). Enlarge rectangle ABCD by scale factor 1/2, centre (2, 4).
(3)
13
Shape T has vertices (1, 1), (1, 3) and (3, 1). Shape U has vertices (1, 1), (1, 7) and (7, 1). Describe fully the single transformation that maps shape T onto shape U.
(3)
14
A shape is translated by the vector (3, -1) and then by the vector (-5, 4). Which single vector describes the combined translation?
A) (2, 3)
B) (-2, 3)
C) (8, -5)
D) (-8, 5)
15
Triangle ABC has vertices A(1, 1), B(3, 1) and C(1, 3). Triangle ABC is enlarged by scale factor -2, centre (0, 0). Find the coordinates of A', B' and C'.
(3)
16
Two rectangles are similar. Rectangle A measures 4 cm by 6 cm. Rectangle B is an enlargement of rectangle A with scale factor 2.5.
Diagram NOT accurately drawn
(a)Work out the dimensions of rectangle B.(2)
(b)Work out the area of rectangle B.(2)
(c)Amir says, 'The area of rectangle B is 2.5 times the area of rectangle A.' Explain why Amir is wrong, and state the correct area scale factor.(2)
17
Triangle ABC is reflected, and the image is then rotated, to form triangle A'B'C'.
(a)State, giving a reason, whether triangle A'B'C' is congruent to triangle ABC.(1)
(b)State, giving a reason, whether triangle A'B'C' is similar to triangle ABC.(1)
18
A map has a scale of 1 : 25000.
(a)The distance between the villages of Ashcombe and Bramerton on the map is 6.4 cm. Work out the real distance between the villages, in kilometres.(3)
(b)A footpath on the map is being redrawn. Its length on the map changes from 3 cm to 4.5 cm, at the same scale. Work out the increase in the real length of the footpath, in metres.(3)
19
A model of a statue is built using a linear scale factor of 1 : 20 (the actual statue is 20 times bigger in each direction than the model). The model has a base area of 15 cm2 and a volume of 45 cm3.
(a)Work out the base area of the actual statue, in m2.(3)
(b)Work out the volume of the actual statue, in m3.(3)
20
Point P has coordinates (a, b). P is reflected in the line y = x, and the image is then reflected in the x-axis, to give point P''.
(4)
Mark scheme · KS3.M-G3 Geometry: Transformations
Question 1
(a) B1 (4, -1) cao
(a) Answer: A' = (4, -1)
(b) B1 (7, -1) cao
(b) Answer: B' = (7, -1)
(c) B1 (4, 1) cao
(c) Answer: C' = (4, 1)
Question 2
(a) B1 (5, -2) cao
(a) Answer: (5, -2)
(b) B1 (-5, 2) cao
(b) Answer: (-5, 2)
Question 3
(a) B1 4 cao
(a) Answer: 4
(b) B1 5 cao
(b) Answer: 5
(c) B1 1 cao (accept 'no rotational symmetry')
(c) Answer: 1 (no rotational symmetry other than 360 degrees)
Question 4
(a) B1 (5, -2) cao
(a) Answer: (5, -2)
(b) B1 (3, -4) cao
(b) Answer: (3, -4)
Question 5
M1 recognises a translation and attempts the vector, e.g. compares one pair of vertices
A1 translation by vector (3, 2) oe (must state 'translation' and give the vector)
Answer: Translation by vector (3, 2)
Question 6
(a) B1 (2, 1) cao
(a) Answer: (2, 1)
(b) B1 (2, 4) cao
(b) Answer: (2, 4)
(c) B1 (5, 1) cao
(c) Answer: (5, 1)
Question 7
M1 multiplies each coordinate by 2, seen for at least one vertex
A1 A'(2, 2), B'(6, 2), C'(6, 4), D'(2, 4) all correct cao
Answer: A'(2, 2), B'(6, 2), C'(6, 4), D'(2, 4)
Question 8
M1 correct method: image = centre + 3 x (vertex - centre), used for at least one vertex
A1 A'(4, 4) correct
A1 B'(10, 4) and C'(4, 10) both correct
Answer: A'(4, 4), B'(10, 4), C'(4, 10)
Question 9
M1 identifies the transformation as a rotation (not a translation or reflection)
A1 angle and direction correct: 90 degrees anticlockwise
A1 centre correct: (0, 0)
Answer: Rotation of 90 degrees anticlockwise about the origin (0, 0)
(a) M1 adds the vectors correctly, e.g. (2+6-2, 3+8+5) oe or finds intermediate point (8, 11)
(a) A1 (6, 16) cao
(a) Answer: (6, 16)
(b) M1 correct displacement vector (4, 13) oe, ft their part (a)
(b) M1 correct use of Pythagoras: √42 + 132
(b) A1 awrt 13.6 m
(b) Answer: 13.6 m (1 dp)
Question 12
M1 correct method: image = centre + 0.5 x (vertex - centre), used for at least one vertex
A1 A'(2, 4) and B'(4, 4) both correct
A1 C'(4, 6) and D'(2, 6) both correct
Answer: A'(2, 4), B'(4, 4), C'(4, 6), D'(2, 6)
Question 13
M1 identifies the invariant point (1, 1) as the centre of enlargement
A1 scale factor 3 found correctly, e.g. from ratio of a pair of corresponding lengths
A1 fully describes: enlargement, scale factor 3, centre (1, 1)
Answer: Enlargement, scale factor 3, centre (1, 1)
Question 14
B1 B cao
Answer: B) (-2, 3)
Question 15
M1 recognises the image lies on the opposite side of the centre from the object (negative scale factor)
M1 correct method: multiplies each coordinate by -2
A1 A'(-2, -2), B'(-6, -2), C'(-2, -6) all correct cao
Answer: A'(-2, -2), B'(-6, -2), C'(-2, -6)
Question 16
(a) M1 4 x 2.5 and 6 x 2.5 oe
(a) A1 10 cm and 15 cm cao
(a) Answer: 10 cm by 15 cm
(b) M1 10 x 15 oe, ft from part (a)
(b) A1 150 cm2 cao
(b) Answer: 150 cm2
(c) M1 compares area of A (24 cm2) with area of B (150 cm2), giving ratio 150/24 = 6.25 oe
(c) A1 correct explanation: the area scale factor is the linear scale factor squared, 2.52 = 6.25, not 2.5
(c) Answer: Amir is wrong; the area scale factor is 2.52 = 6.25, not 2.5
Question 17
(a) B1 yes, congruent, because reflections and rotations are both rigid transformations (isometries) that preserve all side lengths and angles
(a) Answer: Yes, congruent
(b) B1 yes, similar, because congruent shapes are a special case of similar shapes with scale factor 1 (angles equal and sides in the same ratio)
(b) Answer: Yes, similar
Question 18
(a) M1 6.4 x 25000
(a) M1 correct conversion of cm to km (divide by 100000)
(a) A1 1.6 km cao
(a) Answer: 1.6 km
(b) M1 real length before = 3 x 25000 = 75000 cm (= 750 m) and real length after = 4.5 x 25000 = 112500 cm (= 1125 m)
(b) M1 correct conversion of cm to m (divide by 100) for at least one length
(b) A1 375 m cao
(b) Answer: 375 m
Question 19
(a) M1 area scale factor = 202 = 400
(a) M1 15 x 400 = 6000 cm2
(a) A1 0.6 m2 cao (correct conversion, dividing by 10000)
(a) Answer: 0.6 m2
(b) M1 volume scale factor = 203 = 8000
(b) M1 45 x 8000 = 360000 cm3
(b) A1 0.36 m3 cao (correct conversion, dividing by 1000000)
(b) Answer: 0.36 m3
Question 20
M1 reflects (a, b) in y = x correctly to get (b, a)
M1 reflects (b, a) in the x-axis correctly to get (b, -a)
M1 states the rule for a rotation of 90 degrees clockwise about the origin: (x, y) -> (y, -x)
A1 cso: correctly concludes that (a, b) -> (b, -a) matches the rotation rule, so the combined transformation is equivalent to a rotation of 90 degrees clockwise about the origin O
Answer: The combined transformation is equivalent to a rotation of 90 degrees clockwise about the origin O