Dr J Frost (jfrost@tiffin.kingston.sch.uk) GCSE Similarity Dr J Frost (jfrost@tiffin.kingston.sch.uk) GCSE Revision Pack References: 131, 137, 171, 172 Last modified: 19th February 2015

GCSE Specification Pack Ref Description 171 Solve problems involving finding lengths in similar shapes. 172 Understand the effect of enlargement for perimeter, area and volume of shapes and solids. Know the relationships between linear, area and volume scale factors of mathematically similar shapes and solids 131 Convert between units of area 137 Convert between volume measures, including cubic centimetres and cubic metres

Similarity vs Congruence Two shapes are congruent if: ! ? They are the same shape and size (flipping is allowed) Two shapes are similar if: ! ? They are the same shape (flipping is again allowed) b b b a a a

Similarity These two triangles are similar. What is the missing length, and why? 5 ? 7.5 8 12 There’s two ways we could solve this: The ratio of the left side and bottom side is the same in both cases, i.e.: 5 8 = 𝑥 12 Find scale factor: 12 8 Then multiply or divide other sides by scale factor as appropriate. 𝑥=5× 12 8

Quickfire Examples Given that the shapes are similar, find the missing side (the first 3 can be done in your head). 1 2 10 12 ? 32 ? 24 15 18 15 20 4 3 17 24 11 20 40 25 ? 25.88 ? 30

Harder Problems Work out with your neighbour. The diagram shows a square inside a triangle. DEF is a straight line. What is length EF? (Hint: you’ll need to use Pythag at some point) 1 In the diagram BCD is similar to triangle ACE. Work out the length of BD. 2 Since EC = 12cm, by Pythagoras, DC = 9cm. Using similar triangles AEF and CDE: 15 9 = 𝐸𝐹 12 Thus 𝐸𝐹=20 ? 𝐵𝐷 4 = 7.5 10 → 𝐵𝐷=3 ?

(Vote with your diaries) What is the length x? 1 4 x 8 8 9 10 12

(Vote with your diaries) What is the length x? 4 8 9 x 5 6 6.5 6.66...

(Vote with your diaries) What is the length x? 7.5 x 15 10 5 11.25 3 6.5

Exercise 1 7 2𝑐𝑚 1 𝐴 A swimming pool is filled with water. Find 𝑥. 5𝑐𝑚 2 5 3 4 𝑟 3.75 4 3𝑐𝑚 12𝑐𝑚 15𝑚 𝑦 𝑥 12𝑐𝑚 10𝑐𝑚 1.2𝑚 9𝑐𝑚 3.7𝑚 𝐵 𝐶 ? 𝑥=5.25 𝑦=5.6 𝑥 ? 𝑟=3.75𝑐𝑚 ? 𝑩𝑪=𝟖𝒄𝒎 𝑨𝑪=𝟏𝟐.𝟓𝒄𝒎 ? 𝑥=10.8 ? 1.8𝑚 5 6 6 3 [Source: IMC] The diagram shows a square, a diagonal and a line joining a vertex to the midpoint of a side. What is the ratio of area 𝑃 to area 𝑄? N1 N2 8 5 4 5 𝑥 3 𝑥 7 𝑥=4.2 ? 𝑥=1.5 ? 4 [Source: IMO] A square is inscribed in a 3-4-5 right-angled triangle as shown. What is the side-length of the square? N3 Let 𝑎 and 𝑏 be the lengths of the two shorter sides of a right-angled triangle, and let ℎ be the distance from the right angle to the hypotenuse. Prove 1 𝑎 2 + 1 𝑏 2 = 1 ℎ 2 The two unlabelled triangles are similar, with bases in the ratio 2:1. If we made the sides of the square say 6, then the areas of the four triangles are 12, 15, 6, 3. 𝑷:𝑸=𝟔:𝟏𝟓 ? 𝐴 Suppose the length of the square is 𝒙. Then 𝟑−𝒙 𝒙 = 𝒙 𝟒−𝒙 . Solving: 𝒙= 𝟏𝟐 𝟕 ? By similar triangles 𝑨𝑯= 𝒂𝒉 𝒃 Using Pythag on 𝚫𝑨𝑶𝑯: 𝒂 𝟐 = 𝒉 𝟐 + 𝒂 𝟐 𝒉 𝟐 𝒃 𝟐 Divide by 𝒂 𝟐 𝒉 𝟐 and we’re done. 𝐻 ? 𝑎 ℎ 𝑂 𝐵 𝑏

A4/A3/A2 paper 𝑥 “A” sizes of paper (A4, A3, etc.) have the special property that what two sheets of one size paper are put together, the combined sheet is mathematically similar to each individual sheet. What therefore is the ratio of length to width? A5 𝑦 A4 ? 𝑥 𝑦 = 2𝑦 𝑥 ∴ 𝑥= 2 𝑦 So the length is 2 times greater than the width. A5

Scaling areas and volumes A Savvy-Triangle is enlarged by a scale factor of 3 to form a Yusutriangle. 2cm ? 6cm 3cm 9cm ? Area = 3cm2 ? Area = 27cm2 ? Length increased by a factor of 3 ? Area increased by a factor of 9 ?

Scaling areas and volumes For area, the scale factor is squared. For volume, the scale factor is cubed. Example: A shape X is enlarged by a scale factor of 5 to produce a shape Y. The area of shape X is 3m2. What is the area of shape Y? Shape X Shape Y Bro Tip: This is my own way of working out questions like this. You really can’t go wrong with this method! Length: Area: ×5 ×25 ? 3m2 ? 75m2 Example: Shape A is enlarged to form shape B. The surface area of shape A is 30cm2 and the surface area of B is 120cm2. If shape A has length 5cm, what length does shape B have? Shape A Shape B Length: Area: 5cm ? ×2 10cm ? ? ×4 30cm2 120cm2

Scaling areas and volumes For area, the scale factor is squared. For volume, the scale factor is cubed. Example 3: Shape A is enlarged to form shape B. The surface area of shape A is 30cm2 and the surface area of B is 270cm2. If the volume of shape A is 80cm3, what is the volume of shape B? Shape A Shape B ? Length: Area: Volume: ×3 ×9 ? 30cm2 270cm2 ×27 ? 80cm3 2160cm3 ?

B A Test Your Understanding ? Answer = 320cm2 20cm2 320cm2 10cm3 These 3D shapes are mathematically similar. If the surface area of solid A is 20cm2. What is the surface area of solid B? B A Volume = 10cm3 Volume = 640cm3 ? Solid A Solid B Length: Area: Volume: ×4 Answer = 320cm2 ×16 20cm2 320cm2 ×64 10cm3 640cm3

Exercises Copy the table and determine the missing values. [2003] Cylinder A and cylinder B are mathematically similar. The length of cylinder A is 4 cm and the length of cylinder B is 6 cm. The volume of cylinder A is 80cm3. Calculate the volume of cylinder B. 𝟖× 𝟏.𝟓 𝟑 =𝟐𝟕𝟎𝒄 𝒎 𝟑 1 5 Shape A Shape B Length: Area: Volume: 3cm 5cm2 10cm3 ×2 6cm 20cm2 80cm3 ×4 ? ? ? ×8 ? ? 2 Determine the missing values. [2007] Two cones, P and Q, are mathematically similar. The total surface area of cone P is 24cm2. The total surface area of cone Q is 96cm2. The height of cone P is 4 cm. (a) Work out the height of cone Q. 𝟗𝟔÷𝟐𝟒 =𝟐 𝟒×𝟐=𝟖𝒄𝒎 (b) The volume of cone P is 12 cm3. Work out the volume of cone Q. 𝟏𝟐× 𝟐 𝟑 =𝟗𝟔𝒄𝒎𝟑 6 Shape A Shape B Length: Area: Volume: 5m 8m2 12m3 ×3 ? 15m 72m2 324m3 ? ×9 ? ×27 ? ? ? 3 Determine the missing values. Shape A Shape B Length: Area: Volume: 1cm 4cm2 3cm3 ×5 ? 5cm 100cm2 375cm3 ? ? ×25 ? ? ×125 ? 7 The surface area of shapes A and B are 𝑥 and 𝑦 respectively. Given that the length of shape B is 𝑧, write an expression (in terms of 𝑥, 𝑦 and 𝑧) for the length of shape A. 𝒛÷ 𝒚 𝒙 → 𝒛 𝒙 𝒚 4 Determine the missing values. Shape A Shape B Length: Area: Volume: 6m 8m2 10cm3 ×1.5 ? ? 9m 18m2 33.75cm3 ? ×2.25 ? ? ×3.375 ?

Test Your Understanding Bro Hint: Scaling mass is the same as scaling what? Volume ? ? Scale factor of area: 50 18 = 25 9 Scale factor of length: 25 9 = 5 3 Scale factor of volume/mass: 5 3 3 = 125 27 500÷ 125 27 =𝟏𝟎𝟖𝒈

Units of Area and Volume We can use the same principle to find how to convert between units of volume and area. 1m 100cm 1m 100cm 𝑨𝒓𝒆𝒂=𝟏 𝒎 𝟐 ? 𝑨𝒓𝒆𝒂=𝟏𝟎 𝟎𝟎𝟎 𝒎 𝟐 ? Example: What is 8.3m2 in cm2? 8 𝑚 2 83 000 𝑐 𝑚 2 × 100 2 ? ?

Quickfire Questions 1 What is 42cm2 in mm2? 42 𝑐𝑚 2 4 200 𝑚 𝑚 2 5 What is 5.1cm2 in mm2? 5.1 𝑚 2 510𝑚 𝑚 2 × 10 2 ? × 10 2 ? ? ? What is 2m2 in mm2? 2 𝑚 2 2 000 000 𝑚 𝑚 2 What is 2km3 in m3? 2 𝑘𝑚 3 2 000 000 000 𝑚 3 2 6 × 1000 2 ? × 1000 3 ? ? ? What is 3m3 in cm3? 3 𝑚 2 3 000 000 𝑚 𝑚 2 What is 4.25m2 in mm3? 4.25 𝑚 2 4 250 000 000 𝑚 𝑚 2 3 7 × 100 3 ? × 1000 3 ? ? ? 4 What is 13cm3 in mm3? 13 𝑚 2 13 000 𝑚 𝑚 2 8 What is 10.01km2 in mm2? 10.01 𝑘𝑚 2 10 010 000 000 000 𝑚 𝑚 2 × 1000000 2 ? × 10 3 ? ? ?