Shape A has a length of 5 cm. The corresponding length on a mathematically similar shape B is 20 cm.
(Total for Question 1 is 1 mark)
2
Rectangle C measures 6 cm by 4 cm. Rectangle D is mathematically similar to rectangle C, with a scale factor of enlargement of 2.5.
(Total for Question 2 is 2 marks)
3
Triangle ABC is similar to triangle PQR, with AB corresponding to PQ and BC corresponding to QR. AB = 6 cm, PQ = 9 cm and BC = 7 cm.
Diagram NOT accurately drawn
(Total for Question 3 is 2 marks)
4
Two mathematically similar shapes have sides in the ratio 2 : 5. A side on the smaller shape has length 9 cm.
(Total for Question 4 is 2 marks)
5
Trapezium 1 has parallel sides of length 8 cm and 15 cm. Trapezium 2 is mathematically similar to trapezium 1. The side in trapezium 2 corresponding to the 8 cm side has length 5.6 cm.
Diagram NOT accurately drawn
(Total for Question 5 is 2 marks)
6
Rectangle A measures 6 cm by 9 cm.
(a)Which of these rectangles is mathematically similar to rectangle A?(1)
A) 8 cm by 13 cm
B) 12 cm by 18 cm
C) 9 cm by 13 cm
D) 15 cm by 21 cm
(Total for Question 6 is 1 mark)
7
Shape M and shape N are mathematically similar. The scale factor of enlargement from shape M to shape N is 3/4. A side on shape M has length 20 cm.
(Total for Question 7 is 2 marks)
8
Triangle STU is similar to triangle VWX, with ST corresponding to VW and TU corresponding to WX. ST = 5 cm, VW = 15 cm, TU = 8 cm and WX = (3x + 3) cm.
Diagram NOT accurately drawn
(Total for Question 8 is 3 marks)
9
Shape 1 has sides of length 4.5 cm and 6 cm. Shape 2 is mathematically similar to shape 1. The side on shape 2 corresponding to the 4.5 cm side has length 3 cm.
(Total for Question 9 is 2 marks)
10
Deepak measures the shadow of a tent pole of height 1.8 m and finds it is 2.4 m long. At the same moment, a nearby flagpole casts a shadow of length 6 m. The pole, the flagpole and their shadows form similar right-angled triangles.
(Total for Question 10 is 3 marks)
11
Quadrilateral ABCD is mathematically similar to quadrilateral WXYZ, with AB corresponding to WX, BC corresponding to XY, CD corresponding to YZ, and DA corresponding to ZW. AB = 9 cm, WX = 15 cm and CD = 12 cm.
Diagram NOT accurately drawn
(Total for Question 11 is 3 marks)
12
Two mathematically similar shapes have sides in the ratio 3 : 8. A side on the smaller shape has length 12 cm.
(Total for Question 12 is 2 marks)
13
Two right-angled triangles are mathematically similar, with a scale factor of enlargement of 4. The hypotenuse of the smaller triangle is 13 cm.
(Total for Question 13 is 1 mark)
14
A photocopier enlarges a document using a scale factor of 1.4. The original document has a width of 21 cm.
(Total for Question 14 is 2 marks)
15
Shape A has sides of length 4 cm and x cm. Shape B is mathematically similar to shape A, with corresponding sides of length 10 cm and 12 cm (10 cm corresponding to the 4 cm side).
(Total for Question 15 is 3 marks)
16
Two squares are drawn, of any size.
(Total for Question 16 is 1 mark)
17
A shape is enlarged by 150% to create a mathematically similar shape. A side on the original shape has length 8 cm.
(Total for Question 17 is 2 marks)
18
A company logo is a rectangle measuring 8 cm by 5 cm. The logo is enlarged, keeping it mathematically similar, to print on a banner with a width of 96 cm.
(a)Work out the scale factor of the enlargement.(1)
(b)Work out the height of the banner version of the logo.(2)
(Total for Question 18 is 3 marks)
19
Rectangle 1 has dimensions 6 cm by 15 cm. Rectangle 2 has dimensions 8 cm by 20 cm. Leah says, 'Rectangle 1 and rectangle 2 are mathematically similar.'
Diagram NOT accurately drawn
(Total for Question 19 is 3 marks)
20
In triangle ABC, D lies on AB and E lies on AC, such that DE is parallel to BC. AD = 5 cm, DB = 7 cm and DE = 6 cm.
Diagram NOT accurately drawn
(Total for Question 20 is 4 marks)
21
In the diagram, PQ is parallel to ST. The straight lines PT and QS intersect at M. PM = 8 cm, MT = 10 cm and QM = 12 cm.
Diagram NOT accurately drawn
(a)Explain why triangle PMQ is similar to triangle TMS.(1)
(b)Work out the length of MS.(3)
(Total for Question 21 is 4 marks)
22
Shape A is mathematically similar to shape B, with a scale factor of enlargement from A to B of 2. Shape B is mathematically similar to shape C, with a scale factor of enlargement from B to C of 1.5. A side on shape A has length 4 cm.
Diagram NOT accurately drawn
(a)Work out the (compound) scale factor of enlargement from shape A to shape C.(2)
(b)Hence work out the length of the corresponding side on shape C.(2)
(Total for Question 22 is 4 marks)
23
Nadia is designing two mathematically similar flags for a school fete. The small flag has width (2x + 1) cm and height (x + 4) cm. The large flag has width 45 cm and height 40 cm. The scale factor of enlargement from the small flag to the large flag is 5.
Diagram NOT accurately drawn
(a)Using the width, form and solve an equation to find the value of x.(3)
(b)State the width and height of the small flag, and verify your value of x using the height dimension.(2)
(Total for Question 23 is 5 marks)
24
In triangle ABC, D is a point on AC such that angle ADB = angle ABC. AB = 12 cm, AD = x cm and AC = (x + 7) cm.
Diagram NOT accurately drawn
(a)Explain why triangle ABD is similar to triangle ACB.(1)
(b)Show that x2 + 7x - 144 = 0.(2)
(c)Solve the equation to find the value of x, explaining why the other solution is rejected.(2)
(d)Hence state the length of AC.(1)
(Total for Question 24 is 6 marks)
25
A rectangle has width w cm. A mathematically similar, larger rectangle is formed by enlarging the original rectangle by a scale factor of k. The perimeter of the larger rectangle is 3 times the perimeter of the original rectangle.
(Total for Question 25 is 2 marks)
Mark scheme · 5.18D Similar Shapes (Lengths): Fluency and Exam Drill
Question 1
B1 4 cao
Answer: 4
Question 2
M1 6 x 2.5 and 4 x 2.5 oe (both dimensions scaled by 2.5)
A1 15 cm by 10 cm cao
Answer: 15 cm by 10 cm
Question 3
M1 scale factor = 9/6 oe (= 1.5)
A1 10.5 (cm) cao
Answer: 10.5 cm
Question 4
M1 9 x 5/2 oe
A1 22.5 (cm) cao
Answer: 22.5 cm
Question 5
M1 scale factor = 5.6/8 oe (= 0.7)
A1 10.5 (cm) cao
Answer: 10.5 cm
Question 6
(a) B1 B) 12 cm by 18 cm
(a) Answer: B
Question 7
M1 20 x 3/4 oe
A1 15 (cm) cao
Answer: 15 cm
Question 8
M1 scale factor = 15/5 oe (= 3)
M1 3x + 3 = 8 x 3 oe, i.e. 3x + 3 = 24
A1 x = 7 cao
Answer: x = 7
Question 9
M1 scale factor = 3/4.5 oe (= 2/3)
A1 4 (cm) cao
Answer: 4 cm
Question 10
M1 scale factor = 6/2.4 oe (= 2.5)
M1 1.8 x their 2.5
A1 4.5 (m) cao
Answer: 4.5 m
Question 11
M1 scale factor = 15/9 oe (= 5/3)
M1 12 x their 5/3
A1 20 (cm) cao
Answer: 20 cm
Question 12
M1 12 x 8/3 oe
A1 32 (cm) cao
Answer: 32 cm
Question 13
B1 52 (cm) cao
Answer: 52 cm
Question 14
M1 21 x 1.4 oe
A1 29.4 (cm) cao
Answer: 29.4 cm
Question 15
M1 scale factor = 10/4 oe (= 2.5)
M1 their 2.5 x x = 12 oe (correct equation formed)
A1 x = 4.8 cao
Answer: x = 4.8
Question 16
B1 Yes, because all squares have four equal sides and four right angles, so the ratio of corresponding sides is always the same (equal to 1) oe
Answer: Yes, any two squares are mathematically similar.
Question 17
M1 8 x 1.5 oe (150% = scale factor 1.5)
A1 12 (cm) cao
Answer: 12 cm
Question 18
(a) B1 12 cao
(a) Answer: 12
(b) M1 5 x their 12
(b) A1 60 (cm) cao ft their (a)
(b) Answer: 60 cm
Question 19
M1 15/6 (= 2.5) or 6/15 (= 0.4) found
M1 20/8 (= 2.5) or 8/20 (= 0.4) found
A1 correct conclusion: Yes, Leah is correct, because the ratios of corresponding sides are equal (both 2.5), cao
Answer: Yes, Leah is correct: 15/6 = 2.5 and 20/8 = 2.5, so the ratios are equal.
Question 20
M1 AB = 5 + 7 (= 12) (whole side AB found)
M1 scale factor = 5/12 oe (triangle ADE similar to triangle ABC since DE parallel to BC, AA)
M1 BC = 6 / their (5/12) oe, i.e. 6 x 12/5
A1 14.4 (cm) cao
Answer: 14.4 cm
Question 21
(a) B1 angle QPM = angle STM and angle PQM = angle TSM (alternate angles, PQ parallel to ST), and angle PMQ = angle SMT (vertically opposite angles), so the triangles are similar (AA)
(a) Answer: Alternate angles (PQ parallel to ST) and vertically opposite angles at M give two pairs of equal angles, so the triangles are similar (AA).
(b) M1 PM/MT = QM/MS oe (correct ratio equation set up, e.g. 8/10 = 12/MS)
(b) M1 MS = 12 x 10/8 oe (correct rearrangement)
(b) A1 15 (cm) cao
(b) Answer: 15 cm
Question 22
(a) M1 2 x 1.5 oe (compound scale factor)
(a) A1 3 cao
(a) Answer: 3
(b) M1 4 x their 3
(b) A1 12 (cm) cao ft their (a)
(b) Answer: 12 cm
Question 23
(a) M1 5(2x + 1) = 45 oe (correct equation formed from the width)
(a) M1 2x + 1 = 9 oe (correct simplification)
(a) A1 x = 4 cao
(a) Answer: x = 4
(b) B1 width = 9 cm and height = 8 cm cao ft their (a)
(b) B1 valid check shown using the height, e.g. 5(x + 4) = 5 x 8 = 40, matching the large flag's height, confirming x = 4
(b) Answer: Width = 9 cm, height = 8 cm (checks: 9 x 5 = 45 and 8 x 5 = 40)
Question 24
(a) B1 angle A is common to both triangles, and angle ADB = angle ABC (given), so triangle ABD is similar to triangle ACB (AA)
(a) Answer: Angle A is common, and angle ADB = angle ABC is given, so the triangles are similar by AA.
(b) M1 AB/AC = AD/AB oe, i.e. 12/(x + 7) = x/12 (correct ratio from the similar triangles)
(b) C1 correct expansion and rearrangement to x2 + 7x - 144 = 0 cso
(b) Answer: x2 + 7x - 144 = 0 (shown)
(c) M1 (x - 9)(x + 16) = 0 oe, or correct use of the quadratic formula
(c) A1 x = 9, rejecting x = -16 since a length cannot be negative
(c) Answer: x = 9 (x = -16 is rejected as a length cannot be negative)
(d) B1 16 (cm) cao ft their x
(d) Answer: 16 cm
Question 25
B1 k = 3 cao
B1 correct justification: because the perimeter (like any length) of a similar shape scales by the same linear scale factor as its corresponding sides, oe