Edexcel GCSE Maths (1MA1), Higher tier · Geometry and measures
Practise Similar shapes (length, area, volume). 3 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
In triangle PQR, angle P = 40° and angle Q = 85°. In triangle STU, angle T = 55° and angle U = 40°.
(a) Which condition shows that triangle PQR is similar to triangle STU? Tick (✓) one box.[1]
AA (two pairs of equal angles)
SAS (two pairs of sides in the same ratio, and the angles between them equal)
SSS (all three pairs of sides in the same ratio)
None of these: the information does not show that the triangles are similar
(b) Complete this statement so that the letters of the two triangles are in corresponding order. Triangle PQR is similar to triangle ..........[1]
Show the answer and mark scheme
(a)Answer: AA (two pairs of equal angles)
B1 for AA (two pairs of equal angles)
(b)Answer: triangle UST
B1 for UST
Worked solution: Angle R = 180 − 40 − 85 = 55° and angle S = 180 − 55 − 40 = 85°. So angle P = angle U (40°), angle Q = angle S (85°) and angle R = angle T (55°): triangle PQR is similar to triangle UST.
Question 2Medium3 marks
In triangle JKL, JK = 8 cm and JL = 20 cm. In triangle XYZ, XY = 10 cm and XZ = 25 cm. Angle KJL = angle YXZ.
(a) Which condition shows that triangle JKL is similar to triangle XYZ? Tick (✓) one box.[1]
AA (two pairs of equal angles)
SAS (two pairs of sides in the same ratio, and the angles between them equal)
SSS (all three pairs of sides in the same ratio)
None of these: the information does not show that the triangles are similar
(b) KL = 20 cm. Work out the length of YZ.[2]
Show the answer and mark scheme
(a)Answer: SAS (two pairs of sides in the same ratio, and the angles between them equal)
B1 for SAS (two pairs of sides in the same ratio, and the angles between them equal)
(b)Answer: 25 cm
M1 for the scale factor \(\frac{5}{4}\) oe (e.g. 10 ÷ 8)
A1 for 25
Worked solution: \(\frac{XY}{JK} = \frac{10}{8} = \frac{5}{4}\) and \(\frac{XZ}{JL} = \frac{25}{20} = \frac{5}{4}\), and the angles between these sides are equal, so the triangles are similar (SAS). \(YZ = 20 \times \frac{5}{4} = 25\) cm.
Question 3Hard5 marks
Two bottles are mathematically similar. The small bottle has a height of 12 cm and holds 400 ml. The large bottle has a height of 18 cm.
(a) Ian says, 'The large bottle holds 600 ml, because 18 cm is 1.5 times 12 cm.' Explain why Ian is wrong.[1]
(b) The small bottle costs £1.60. The large bottle costs £4.80. Which bottle is better value for money? You must show your working.[4]
Show the answer and mark scheme
(a)Answer: Volumes scale by the cube of the length scale factor: \(1.5^3 = 3.375\), not 1.5.
C1 for explaining that the volume scale factor is \(1.5^3\) (the cube of the length scale factor), not 1.5
Worked solution: Every length is multiplied by 1.5, so the volume is multiplied by \(1.5 \times 1.5 \times 1.5 = 3.375\).
(b)Answer: Large: it holds 1350 ml, so 281.25 ml per £1, compared with 250 ml per £1 for the small bottle.
P1 for \(1.5^3\) (= 3.375)
P1 for \(400 \times 3.375 = 1350\) (ml)
P1 for a correct comparison, e.g. \(400 \div 1.60 = 250\) and \(1350 \div 4.80 = 281.25\) ml per £, or 0.4p and 0.356p per ml, or price ratio 3 compared with volume ratio 3.375
C1 for the large bottle, supported by correct figures
Worked solution: Large bottle \(= 400 \times 1.5^3 = 400 \times 3.375 = 1350\) ml. Small: \(400 \div 1.60 = 250\) ml per £1. Large: \(1350 \div 4.80 = 281.25\) ml per £1. The large bottle gives more per pound, so it is better value.