A shape is enlarged using a negative scale factor. Which of these correctly describes the image compared with the object?
A) Same side of the centre of enlargement, same orientation
B) Same side of the centre of enlargement, but rotated 180 degrees (upside down)
C) Opposite side of the centre of enlargement, and rotated 180 degrees (upside down)
D) Opposite side of the centre of enlargement, but the same orientation as the object
(Total for Question 1 is 1 mark)
2
State two facts that are true about the image formed when a shape is enlarged using a negative scale factor (compared with the original shape).
(Total for Question 2 is 2 marks)
3
Triangle ABC has vertices A(2,1), B(4,1) and C(2,3). Triangle ABC is enlarged by scale factor -1, centre (0,0), to give triangle A'B'C'. Find the coordinates of A', B' and C'.
(Total for Question 3 is 3 marks)
4
Triangle PQR has vertices P(1,2), Q(3,2) and R(1,5). It is enlarged by scale factor -2, centre (0,0), to give triangle P'Q'R'. Find the coordinates of P', Q' and R'.
(Total for Question 4 is 3 marks)
5
Triangle ABC has vertices A(3,4), B(5,4) and C(3,6). Triangle ABC is enlarged by scale factor -3, centre (1,2), to give triangle A'B'C'. Find the coordinates of A', B' and C'.
(Total for Question 5 is 4 marks)
6
Triangle ABC has vertices A(1,1), B(3,1) and C(1,2). It is enlarged to form triangle A'B'C' with vertices A'(5,-7), B'(-1,-7) and C'(5,-10). Find the scale factor of the enlargement and the coordinates of the centre of enlargement.
(Total for Question 6 is 4 marks)
7
Triangle DEF has vertices D(2,5), E(6,5) and F(2,9). Triangle D'E'F' has vertices D'(2,2), E'(0,2) and F'(2,0). Describe fully the single transformation that maps triangle DEF onto triangle D'E'F'.
(Total for Question 7 is 3 marks)
8
Point A has coordinates (4,6). Point A is enlarged by scale factor -2, centre (1,2), to give point A'. Work out the coordinates of A'.
(Total for Question 8 is 2 marks)
9
Shape P has an area of 8 cm2. Shape P is enlarged by scale factor -2 to form shape Q. Work out the area of shape Q.
(Total for Question 9 is 2 marks)
10
A shape is enlarged by scale factor -3 to form an image. A side on the image has length 24 cm. Work out the length of the corresponding side on the original shape.
(Total for Question 10 is 2 marks)
11
Show that enlarging a shape by scale factor -1 about a centre C has exactly the same effect as rotating the shape by 180 degrees about the same centre C.
(Total for Question 11 is 3 marks)
12
A triangle is enlarged by scale factor -4, centre (0,0). An object vertex is at (2,-1). What are the coordinates of the image of this vertex?
A) (8,-4)
B) (-8,4)
C) (-8,-4)
D) (8,4)
(Total for Question 12 is 1 mark)
13
A square WXYZ has vertices W(1,1), X(3,1), Y(3,3) and Z(1,3). The square is enlarged by scale factor -2, using the centre of the square, (2,2), as the centre of enlargement, to form square W'X'Y'Z'.
(a)Find the coordinates of W', X', Y' and Z'.(4)
(b)The area of square WXYZ is 4 cm2. Use the scale factor of the enlargement to work out the area of square W'X'Y'Z'. Give a reason for your answer.(2)
(Total for Question 13 is 6 marks)
14
Shape A is enlarged by scale factor -3, centre (0,0), to form shape B. Shape B is then enlarged by scale factor -1/3, centre (0,0), to form shape C. Describe fully the single transformation that maps shape A directly onto shape C.
(Total for Question 14 is 3 marks)
15
A drawing tool enlarges shapes by a scale factor of -2.5 about a fixed point. A line segment on the original drawing has length 6 cm. Work out the length of the corresponding line segment on the enlarged copy.
(Total for Question 15 is 2 marks)
16
A rectangle ABCD has vertices A(-2,4), B(2,4), C(2,0) and D(-2,0). The rectangle is enlarged by scale factor -1/2, using centre (0,2), to form rectangle A'B'C'D'.
(a)Find the coordinates of A', B', C' and D'.(4)
(b)The area of rectangle ABCD is 16 cm2. Use the scale factor to calculate the area of rectangle A'B'C'D'.(2)
(c)Find the ratio of the perimeter of ABCD to the perimeter of A'B'C'D', giving your answer in its simplest form.(2)
(Total for Question 16 is 8 marks)
17
A student is asked to enlarge a triangle by scale factor -2 from a given centre. The student draws the image on the same side of the centre as the original triangle, but twice the size. Explain the error in the student's diagram.
(Total for Question 17 is 1 mark)
18
A shape is enlarged by scale factor -0.4. State two facts about the size and position of the image compared with the object.
(Total for Question 18 is 2 marks)
19
A shape is enlarged by scale factor -1. Sunita says, 'The image will have the same area as the original shape.' Is Sunita correct? Justify your answer.
(Total for Question 19 is 2 marks)
20
Triangle A has an area of 18 cm2. It is enlarged by a negative scale factor to form triangle B, which has an area of 72 cm2. Find the scale factor of the enlargement.
(Total for Question 20 is 3 marks)
Mark scheme · 6.9 Enlarging with Negative Scale Factors
Question 1
B1 C oe
Answer: C
Question 2
B1 the image lies on the opposite side of the centre of enlargement to the object oe
B1 the image is inverted / rotated 180 degrees relative to the object (upside down) oe
Answer: The image is on the opposite side of the centre of enlargement to the object, and it is inverted (rotated 180 degrees) compared with the object.
Question 3
B1 A'(-2,-1)
B1 B'(-4,-1)
B1 C'(-2,-3) cao
Answer: A'(-2,-1), B'(-4,-1), C'(-2,-3)
Question 4
B1 P'(-2,-4)
B1 Q'(-6,-4)
B1 R'(-2,-10) cao
Answer: P'(-2,-4), Q'(-6,-4), R'(-2,-10)
Question 5
M1 correct vector method used for at least one vertex, e.g. (A - centre) x -3, then + centre
A1 A'(-5,-4)
A1 B'(-11,-4)
A1 C'(-5,-10) cao
Answer: A'(-5,-4), B'(-11,-4), C'(-5,-10)
Question 6
M1 correct comparison of corresponding lengths, e.g. AB = 2, A'B' = 6, so ratio = 3
A1 scale factor = -3 (negative because image is inverted / opposite orientation)
M1 correct equation set up using one vertex and its image with k = -3 to find the centre (p,q)
A1 centre = (2,-1) cao
Answer: Scale factor = -3; centre of enlargement = (2,-1)
Question 7
B1 enlargement
B1 scale factor -1/2 oe -0.5
B1 centre (2,3) oe
Answer: Enlargement, scale factor -1/2, centre (2,3)
Question 8
M1 vector from centre to A found or equivalent method, e.g. (3,4)
A1 A'(-5,-6) cao
Answer: A'(-5,-6)
Question 9
M1 area scale factor = (-2)2 = 4
A1 32 cm2 cao
Answer: 32 cm2
Question 10
M1 24 / 3 with 3 identified as the magnitude of the scale factor
A1 8 cm cao
Answer: 8 cm
Question 11
M1 correct general image formula for enlargement with k = -1: P' = C + (-1)(P - C) = 2C - P
M1 correct general image formula for a 180 degree rotation about C: P' = 2C - P
A1 both expressions are identical, so the two transformations are the same cso
Answer: Both transformations map any point P to 2C - P, so an enlargement of scale factor -1 about C is identical to a rotation of 180 degrees about C.
Question 12
B1 B
Answer: B
Question 13
(a) B1 W'(4,4)
(a) B1 X'(0,4)
(a) B1 Y'(0,0)
(a) B1 Z'(4,0) cao
(a) Answer: W'(4,4), X'(0,4), Y'(0,0), Z'(4,0)
(b) M1 area scale factor = (-2)2 = 4
(b) A1 16 cm2 cao, with reason that area scale factor is the square of the length scale factor
(b) Answer: 16 cm2
Question 14
B1 overall scale factor = (-3) x (-1/3) = 1
B1 a scale factor of 1 leaves every point unchanged
B1 identity transformation stated, i.e. shape C is identical to shape A in the same position oe
Answer: The combined effect is an enlargement of scale factor 1, so shape C coincides exactly with shape A (identity transformation, no change).
Question 15
M1 6 x 2.5 with 2.5 identified as the magnitude of the scale factor
A1 15 cm cao
Answer: 15 cm
Question 16
(a) B1 A'(1,1)
(a) B1 B'(-1,1)
(a) B1 C'(-1,3)
(a) B1 D'(1,3) cao
(a) Answer: A'(1,1), B'(-1,1), C'(-1,3), D'(1,3)
(b) M1 area scale factor = (-1/2)2 = 1/4
(b) A1 4 cm2 cao
(b) Answer: 4 cm2
(c) M1 perimeter of ABCD = 16, perimeter of A'B'C'D' = 8 (or use scale factor magnitude 1/2 directly)
(c) A1 2 : 1 cao
(c) Answer: 2 : 1
Question 17
B1 the image should be drawn on the opposite side of the centre of enlargement (and inverted), because the scale factor is negative oe
Answer: The image should be on the opposite side of the centre of enlargement (and inverted), because the scale factor is negative.
Question 18
B1 the image is smaller than the object, since the magnitude of the scale factor is less than 1 oe
B1 the image is on the opposite side of the centre and inverted, since the scale factor is negative oe
Answer: The image is smaller than the object (since |scale factor| < 1), and it lies on the opposite side of the centre of enlargement, inverted compared with the object.
Question 19
B1 yes, Sunita is correct
B1 reason: area scale factor = (-1)2 = 1, so the area is unchanged oe
Answer: Yes, Sunita is correct.
Question 20
M1 area ratio = 72 / 18 = 4
M1 k2 = 4, so k = 2 or k = -2
A1 k = -2, since the scale factor is given as negative cao