A shape is enlarged using a negative scale factor. Describe the position of the image in relation to the object and the centre of enlargement.
(Total for Question 1 is 1 mark)
2
A shape is enlarged by scale factor -4. State (i) the effect on the length of each side of the shape (ii) the effect on the orientation of the shape.
(Total for Question 2 is 2 marks)
3
The point P has coordinates (2, 3). P is enlarged by scale factor -1, centre the origin O, to give the image P'. Find the coordinates of P'.
(Total for Question 3 is 2 marks)
4
The point Q has coordinates (4, -2). Q is enlarged by scale factor -2, centre the origin O, to give the image Q'. Find the coordinates of Q'.
(Total for Question 4 is 2 marks)
5
The point R has coordinates (3, 1). R is enlarged by scale factor -3, centre (1, 1), to give the image R'. Find the coordinates of R'.
(Total for Question 5 is 3 marks)
6
Triangle DEF has vertices D(2, 1), E(5, 1) and F(2, 3). The triangle is enlarged by scale factor -1/2, centre the origin O, to give triangle D'E'F'. Find the coordinates of D', E' and F'.
(Total for Question 6 is 3 marks)
7
A shape is enlarged by scale factor -5. State the area scale factor of the enlargement.
(Total for Question 7 is 1 mark)
8
A shape has area 8 cm2. The shape is enlarged by scale factor -2. Work out the area of the image.
(Total for Question 8 is 2 marks)
9
An object has area 6 cm2. After an enlargement with a negative scale factor, its image has area 54 cm2. Find the scale factor of the enlargement.
(Total for Question 9 is 2 marks)
10
Triangle ABC has vertices A(1, 1), B(4, 1) and C(1, 3). After an enlargement, the image has vertices A'(-2, -2), B'(-8, -2) and C'(-2, -6). Find (i) the scale factor of the enlargement (ii) the coordinates of the centre of enlargement.
(Total for Question 10 is 4 marks)
11
A single transformation maps a triangle with vertices (2, 2), (4, 2) and (2, 5) onto a triangle with vertices (-1, -1), (-2, -1) and (-1, -2.5). State (i) the type of transformation (ii) the scale factor.
(Total for Question 11 is 2 marks)
12
Point S has coordinates (2, 2) and maps to the image point S'(-2, -2) under an enlargement with scale factor -1. Find the coordinates of the centre of enlargement.
(Total for Question 12 is 2 marks)
13
Describe fully the single transformation that maps triangle XYZ, with vertices X(2, 1), Y(4, 1) and Z(2, 4), onto triangle X'Y'Z', with vertices X'(-4, -2), Y'(-8, -2) and Z'(-4, -8).
(Total for Question 13 is 3 marks)
14
An enlargement maps a line segment of length 6 cm to an image of length 15 cm. The image is inverted compared with the object. State (i) the scale factor of the enlargement, including its sign (ii) the area scale factor that this would give for a 2-dimensional shape.
(Total for Question 14 is 2 marks)
15
Triangle PQR has vertices P(0, 1), Q(3, 1) and R(0, 4). The area of triangle PQR is 4.5 cm2. Triangle PQR is enlarged by scale factor -2, centre the origin O, to give triangle P'Q'R'.
(a)Find the coordinates of P', Q' and R'.(2)
(b)Find the area of triangle P'Q'R'.(2)
(Total for Question 15 is 4 marks)
16
A student called Aaliyah enlarges a triangle with vertices (2, 0), (4, 0) and (2, 3) by scale factor -2, centre the origin O. Aaliyah's image has vertices (4, 0), (8, 0) and (4, 6).
(a)Explain the error Aaliyah has made.(1)
(b)Write down the correct coordinates of the image vertices.(2)
(Total for Question 16 is 3 marks)
17
Triangle LMN has vertices L(2, 3), M(6, 3) and N(2, t), where t is a positive constant. The triangle is enlarged by scale factor -1/2, centre the origin O, to give triangle L'M'N'.
(a)Find the coordinates of L' and M'.(2)
(b)Given that N' has coordinates (-1, -4), find the value of t.(3)
(Total for Question 17 is 5 marks)
18
An object has area 12 cm2. After an enlargement, the image has area 300 cm2. The scale factor of the enlargement is known to be negative. Find the scale factor, showing your working.
(Total for Question 18 is 3 marks)
19
Declan is investigating how a pinhole camera works for a physics project. Light from an object passes through a small pinhole and forms an inverted image on a screen on the other side of the hole, with the object, the hole and the image point in a straight line. This means the image can be modelled as an enlargement of the object, with the pinhole as the centre of enlargement and a negative scale factor. In Declan's model, the screen is 15 cm from the pinhole, and a toy of height 40 cm stands upright 60 cm from the pinhole on the other side.
(a)Find the scale factor of the enlargement, giving your answer as a fraction.(2)
(b)Calculate the height of the image formed on the screen.(2)
(c)Declan then moves the toy so that it is 90 cm from the pinhole, keeping the screen 15 cm from the pinhole. Calculate the new height of the image formed on the screen.(3)
(Total for Question 19 is 7 marks)
20
Rectangle WXYZ has vertices W(-1, -1), X(3, -1), Y(3, 1) and Z(-1, 1), and has area 8 square units. The rectangle is enlarged by scale factor -3, centre the point (1, 1), to give rectangle W'X'Y'Z'.
(a)Show that W' has coordinates (7, 7), and find the coordinates of X'.(3)
(b)Find the area of rectangle W'X'Y'Z'.(2)
(Total for Question 20 is 5 marks)
21
Triangle ABC has vertices A(a, 2), B(a + 4, 2) and C(a, 5), where a is a constant. The triangle is enlarged by scale factor -1, centre the point (3, 4), to give triangle A'B'C'.
(a)Show that A' has coordinates (6 - a, 6).(3)
(b)Given that A' has coordinates (2, 6), find the value of a.(2)
(Total for Question 21 is 5 marks)
22
Triangle ABC has vertices A(2, 1), B(4, 1) and C(2, 6), and has area 5 square units. Triangle ABC is enlarged to give triangle A'B'C', where A' has coordinates (2, 11), the area of triangle A'B'C' is 80 square units, and the image is inverted compared with the object.
(a)Show that the linear scale factor of the enlargement is -4.(2)
(b)Find the coordinates of the centre of enlargement.(3)
(c)Find the coordinates of C', the image of C.(2)
(Total for Question 22 is 7 marks)
Mark scheme · 6.9D Enlarging with Negative Scale Factors: Fluency and Exam Drill
Question 1
B1 image is on the opposite side of the centre of enlargement to the object (in line with the centre and the object) oe
Answer: The image lies on the opposite side of the centre of enlargement from the object, in line with the centre.
Question 2
B1 each side is 4 times as long (the image is 4 times the size) oe
B1 the image is rotated 180 degrees relative to the object (upside down and back-to-front) oe
Answer: (i) Each side becomes 4 times as long. (ii) The image is rotated 180 degrees compared with the object.
Question 3
M1 P' = -1 * (2, 3) oe, or correct use of the enlargement formula
A1 (-2, -3) cao
Answer: (-2, -3)
Question 4
M1 Q' = -2 * (4, -2) oe
A1 (-8, 4) cao
Answer: (-8, 4)
Question 5
M1 finds the vector from the centre to R: (2, 0) oe
M1 multiplies the vector by -3: (-6, 0) oe
A1 R' = (-5, 1) cao
Answer: (-5, 1)
Question 6
M1 correct method applied to at least one vertex, e.g. D' = -1/2 * (2, 1)
A1 two of D'(-1,-0.5), E'(-2.5,-0.5), F'(-1,-1.5) correct
A1 all three vertices correct cao
Answer: D' = (-1, -0.5); E' = (-2.5, -0.5); F' = (-1, -1.5)
Question 7
B1 25 cao
Answer: 25
Question 8
M1 area scale factor = (-2)2 = 4
A1 32 cm2 cao
Answer: 32 cm2
Question 9
M1 area scale factor = 54/6 = 9, so k2 = 9 oe
A1 k = -3 cao (rejects +3)
Answer: -3
Question 10
M1 compares corresponding vectors, e.g. AB = (3,0) and A'B' = (-6,0)
A1 scale factor = -2 cao
M1 uses a correct method to find the centre, e.g. centre = (A' - kA)/(1-k)
A1 centre = (0, 0) cao
Answer: scale factor = -2; centre of enlargement = (0, 0)
Question 11
B1 enlargement (centre the origin) oe
B1 scale factor = -1/2 (or -0.5) cao
Answer: Enlargement, centre (0, 0), scale factor -1/2
Question 12
M1 recognises that the centre is the midpoint of S and S' when k = -1 oe
A1 (0, 0) cao
Answer: (0, 0)
Question 13
B1 enlargement
B1 scale factor -2
B1 centre (0, 0) oe
Answer: Enlargement, scale factor -2, centre (0, 0)
Question 14
B1 k = -2.5 (or -5/2) cao
B1 area scale factor = 6.25 (or 25/4) cao
Answer: k = -2.5; area scale factor = 6.25
Question 15
(a) M1 multiplies each vertex by -2
(a) A1 P'(0,-2), Q'(-6,-2) and R'(0,-8) all correct cao
(a) B1 states Aaliyah has used scale factor +2 (ignored the negative sign), placing the image on the same side as the object instead of the opposite side oe
(a) Answer: Aaliyah used scale factor +2 instead of -2, so her image is on the same side of the centre as the object instead of the opposite side.
(b) M1 multiplies each vertex by -2
(b) A1 (-4,0), (-8,0) and (-4,-6) all correct cao
(b) Answer: (-4, 0), (-8, 0) and (-4, -6)
Question 17
(a) M1 multiplies each vertex by -1/2
(a) A1 L'(-1,-1.5) and M'(-3,-1.5) both correct cao
(a) Answer: L' = (-1, -1.5); M' = (-3, -1.5)
(b) M1 sets up N' = -1/2 * (2, t) oe
(b) M1 forms the equation -t/2 = -4 oe
(b) A1 t = 8 cao
(b) Answer: t = 8
Question 18
M1 area scale factor = 300/12 = 25
M1 k2 = 25, so k = 5 or k = -5 oe
A1 k = -5 cao (since the scale factor is negative, +5 is rejected)
Answer: -5
Question 19
(a) M1 scale factor magnitude = 15/60 (= 1/4) oe
(a) A1 k = -1/4 cao (negative since the image is inverted)