(a)Which of these rectangles is mathematically similar to rectangle A?(1)
A) 8 cm by 11 cm
B) 15 cm by 24 cm
C) 10 cm by 14 cm
D) 20 cm by 30 cm
(b)Write down the scale factor of enlargement from rectangle A to the rectangle you chose in part (a).(1)
(Total for Question 1 is 2 marks)
2
Two rectangular photo frames are mathematically similar. Frame P measures 6 cm by 9 cm. Frame Q is similar to frame P and has a shorter side of length 10 cm.
Diagram NOT accurately drawn
(Total for Question 2 is 3 marks)
3
Triangle ABC is similar to triangle XYZ, with AB corresponding to XY and BC corresponding to YZ. AB = 8 cm, XY = 12 cm and BC = 10 cm.
Diagram NOT accurately drawn
(Total for Question 3 is 3 marks)
4
Trapezium 1 has parallel sides of length 12 cm and 18 cm. Trapezium 2 is mathematically similar to trapezium 1. The side in trapezium 2 corresponding to the 18 cm side has length 9 cm.
Diagram NOT accurately drawn
(Total for Question 4 is 3 marks)
5
A map is drawn to a scale of 1 : 25000.
(a)The distance between two villages, Little Horden and Upper Sowerby, on the map is 8 cm. Work out the real distance between the villages. Give your answer in kilometres.(2)
(b)A straight road has a real length of 3 km. Work out the length of this road as it would be shown on the map. Give your answer in centimetres.(2)
(Total for Question 5 is 4 marks)
6
In triangle ABC, D lies on AB and E lies on AC, such that DE is parallel to BC. AD = 4 cm, DB = 6 cm and BC = 15 cm.
Diagram NOT accurately drawn
(Total for Question 6 is 4 marks)
7
Quadrilateral ABCD is mathematically similar to quadrilateral PQRS, with AB corresponding to PQ, BC corresponding to QR, CD corresponding to RS, and DA corresponding to SP. AB = 6 cm and PQ = 15 cm.
Diagram NOT accurately drawn
(a)BC = 10 cm. Work out the length of QR.(2)
(b)RS = 21 cm. Work out the length of CD.(2)
(Total for Question 7 is 4 marks)
8
A model car is made using a scale of 1 : 24. The real car has a length of 4.8 m.
(Total for Question 8 is 3 marks)
9
Triangle ABC is similar to triangle DEF, with AB corresponding to DE and BC corresponding to EF. AB = 8 cm, DE = 12 cm, BC = (x + 2) cm and EF = (2x - 1) cm.
Diagram NOT accurately drawn
(a)Work out the value of x.(3)
(b)Hence work out the length of BC.(2)
(Total for Question 9 is 5 marks)
10
In the diagram, AB is parallel to DC. The straight lines AC and BD intersect at X. AX = 6 cm, XC = 9 cm and BX = 8 cm.
Diagram NOT accurately drawn
(a)Explain why triangle AXB is similar to triangle CXD.(1)
(b)Work out the length of XD.(3)
(Total for Question 10 is 4 marks)
11
A photograph measuring 10 cm by 15 cm is enlarged so that it fits exactly into a frame with a longer side of 45 cm. The enlarged photograph is mathematically similar to the original.
(a)Work out the scale factor of the enlargement.(1)
(b)Work out the length of the shorter side of the enlarged photograph.(2)
(Total for Question 11 is 3 marks)
12
Pentagon J and pentagon K are mathematically similar. The perimeter of pentagon J is 42 cm. The scale factor of enlargement from pentagon J to pentagon K is 2.5.
(a)Work out the perimeter of pentagon K.(2)
(b)One side of pentagon J has length 6 cm. Work out the length of the corresponding side of pentagon K.(2)
(Total for Question 12 is 4 marks)
13
Rectangle M has dimensions 4 cm by 7 cm. Rectangle N has dimensions 9 cm by 15 cm. Priya says, 'Rectangle N is an enlargement of rectangle M, so the two rectangles are mathematically similar.'
(Total for Question 13 is 3 marks)
14
A vertical stick of height 1.2 m casts a shadow 2 m long. At the same time, a nearby vertical flagpole casts a shadow 12 m long. The stick and its shadow, and the flagpole and its shadow, can be modelled as two similar right-angled triangles.
Diagram NOT accurately drawn
(a)Explain why the two triangles are similar.(1)
(b)Work out the height of the flagpole. Give your answer in metres.(3)
(Total for Question 14 is 4 marks)
15
Shape X and shape Y are mathematically similar. The scale factor of enlargement from shape X to shape Y is 3/5.
(a)A side of shape X has length 20 cm. Work out the length of the corresponding side of shape Y.(2)
(b)A different side of shape Y has length 21 cm. Work out the length of the corresponding side of shape X.(2)
(Total for Question 15 is 4 marks)
16
A scale drawing of a garden is made. A path in the garden has a real length of 18 m. On the scale drawing, the path is drawn with a length of 6 cm.
(Total for Question 16 is 3 marks)
17
Triangle LMN is similar to triangle PQR, with LM corresponding to PQ and MN corresponding to QR. LM : PQ = 3 : 7. LM = 9 cm and MN = 15 cm.
Diagram NOT accurately drawn
(a)Work out the length of PQ.(2)
(b)Work out the length of QR.(2)
(c)The perimeter of triangle PQR is 84 cm. Work out the perimeter of triangle LMN.(2)
(Total for Question 17 is 6 marks)
18
Rectangle A has sides of length 10 cm and (3x - 1) cm. Rectangle B is mathematically similar to rectangle A. The side of rectangle B corresponding to the 10 cm side has length 25 cm. The side of rectangle B corresponding to the (3x - 1) cm side has length (6x + 2) cm.
Diagram NOT accurately drawn
(a)Work out the value of x.(3)
(b)Hence work out the perimeter of rectangle A.(2)
(Total for Question 18 is 5 marks)
19
Three mathematically similar triangles, A, B and C, are such that triangle B is an enlargement of triangle A with scale factor 1.5, and triangle C is an enlargement of triangle B with scale factor 2. The shortest side of triangle A is 5 cm.
(a)Work out the length of the shortest side of triangle B.(1)
(b)Work out the length of the shortest side of triangle C.(2)
(c)Work out the overall scale factor of enlargement from triangle A to triangle C.(2)
(Total for Question 19 is 5 marks)
20
In the diagram, D lies on AC of triangle ABC, such that angle ABD = angle ACB. AB = 6 cm, AD = 4 cm and AC = 9 cm.
Diagram NOT accurately drawn
(a)Prove that triangle ABD is similar to triangle ACB.(2)
(b)Given that BC = 10.5 cm, work out the length of BD.(3)
(Total for Question 20 is 5 marks)
Mark scheme · 5.18 Similar Shapes (Lengths)
Question 1
(a) B1 B) 15 cm by 24 cm
(a) Answer: B
(b) B1 3 cao ft their (a)
(b) Answer: 3
Question 2
M1 scale factor = 10/6 oe (= 5/3)
M1 9 x their scale factor
A1 15 (cm) cao
Answer: 15 cm
Question 3
M1 scale factor = 12/8 oe (= 1.5 or 3/2)
M1 10 x their scale factor
A1 15 (cm) cao
Answer: 15 cm
Question 4
M1 scale factor = 9/18 oe (= 0.5 or 1/2)
M1 12 x their scale factor
A1 6 (cm) cao
Answer: 6 cm
Question 5
(a) M1 8 x 25000 (= 200000) oe (real distance found in cm)
(a) A1 2 (km) cao
(a) Answer: 2 km
(b) M1 3 km converted to 300000 cm oe
(b) A1 12 (cm) cao
(b) Answer: 12 cm
Question 6
M1 AB = 4 + 6 (= 10) (whole side AB found)
M1 scale factor = 4/10 oe (= 0.4 or 2/5) (triangle ADE similar to triangle ABC since DE parallel to BC, AA)
M1 15 x their scale factor
A1 6 (cm) cao
Answer: 6 cm
Question 7
(a) M1 scale factor = 15/6 oe (= 2.5 or 5/2)
(a) A1 25 (cm) cao
(a) Answer: 25 cm
(b) M1 21 / 2.5 oe (dividing by the scale factor)
(b) A1 8.4 (cm) cao
(b) Answer: 8.4 cm
Question 8
M1 4.8 m converted to 480 cm oe
M1 480 / 24
A1 20 (cm) cao
Answer: 20 cm
Question 9
(a) M1 scale factor = 12/8 oe (= 1.5)
(a) M1 2x - 1 = 1.5(x + 2) oe (correct equation formed and expanded, e.g. 2x - 1 = 1.5x + 3)
(a) A1 x = 8 cao
(a) Answer: x = 8
(b) M1 substitutes their x into (x + 2)
(b) A1 10 (cm) cao ft their (a)
(b) Answer: 10 cm
Question 10
(a) B1 angle XAB = angle XCD and angle XBA = angle XDC (alternate angles, AB parallel to DC), and angle AXB = angle CXD (vertically opposite angles), so the triangles are similar (AA)
(a) Answer: Alternate angles (AB parallel to DC) and vertically opposite angles at X give two pairs of equal angles, so the triangles are similar (AA).
(b) M1 AX/XC = BX/XD oe (correct ratio equation set up, e.g. 6/9 = 8/XD)
(b) M1 XD = 8 x 9/6 oe (correct rearrangement)
(b) A1 12 (cm) cao
(b) Answer: 12 cm
Question 11
(a) B1 3 cao
(a) Answer: 3
(b) M1 10 x their scale factor
(b) A1 30 (cm) cao ft their (a)
(b) Answer: 30 cm
Question 12
(a) M1 42 x 2.5
(a) A1 105 (cm) cao
(a) Answer: 105 cm
(b) M1 6 x 2.5
(b) A1 15 (cm) cao
(b) Answer: 15 cm
Question 13
M1 9/4 (= 2.25) or 4/9 (= 0.4444...) found
M1 15/7 (awrt 2.14) or 7/15 (awrt 0.467) found
A1 correct conclusion: No, Priya is not correct, because the ratios of corresponding sides are not equal (2.25 does not equal awrt 2.14), cao
Answer: No, Priya is not correct: 9/4 = 2.25 but 15/7 = awrt 2.14, so the ratios are not equal.
Question 14
(a) B1 both triangles have a right angle (object vertical to horizontal ground) and the same angle of elevation of the sun (parallel light rays), so they are similar (AA)
(a) Answer: Both triangles have a right angle and the same angle of elevation of the sun, so they are similar (AA).
(b) M1 scale factor = 12/2 oe (= 6)
(b) M1 1.2 x their scale factor
(b) A1 7.2 (m) cao
(b) Answer: 7.2 m
Question 15
(a) M1 20 x 3/5
(a) A1 12 (cm) cao
(a) Answer: 12 cm
(b) M1 21 divide by 3/5 oe (= 21 x 5/3)
(b) A1 35 (cm) cao
(b) Answer: 35 cm
Question 16
M1 18 m converted to 1800 cm oe
M1 1800 / 6 (= 300)
A1 1 : 300 cao
Answer: 1 : 300
Question 17
(a) M1 9 / 3 (= 3) (value of one ratio unit found), then x 7
(a) A1 21 (cm) cao
(a) Answer: 21 cm
(b) M1 15 x 7/3 oe (using the same scale factor as LM to PQ)
(b) A1 35 (cm) cao
(b) Answer: 35 cm
(c) M1 84 x 3/7 oe (dividing by the scale factor 7/3)
(c) A1 36 (cm) cao
(c) Answer: 36 cm
Question 18
(a) M1 scale factor = 25/10 oe (= 2.5)
(a) M1 6x + 2 = 2.5(3x - 1) oe (correct equation formed and expanded, e.g. 6x + 2 = 7.5x - 2.5)
(a) A1 x = 3 cao
(a) Answer: x = 3
(b) M1 3x - 1 = 8 (cm) found, and 2(10 + 8) oe
(b) A1 36 (cm) cao ft their (a)
(b) Answer: 36 cm
Question 19
(a) B1 7.5 (cm) cao
(a) Answer: 7.5 cm
(b) M1 their (a) x 2 oe, or 5 x (1.5 x 2)
(b) A1 15 (cm) cao
(b) Answer: 15 cm
(c) M1 1.5 x 2
(c) A1 3 cao
(c) Answer: 3
Question 20
(a) B1 angle BAD = angle CAB (angle A is common to both triangles)
(a) B1 angle ABD = angle ACB (given), so triangle ABD is similar to triangle ACB (AA)
(a) Answer: Angle A is common, and angle ABD = angle ACB (given), so the triangles are similar by AA.