GCSE Ratio, Scale and Similar Triangles Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 22nd February 2014

RECAP: Ratio ? ? ? Most ratio problems come in three flavours: 1 Total given. 3 Difference given The ratio of cats to dogs in Battersea Dogs Home is 3:5. There are 240 animals in total. How many cats are there? Some pocket money is shared between Alice and Bob in the ratio 3:7. Bob gets £16 more than Alice. What was the total amount of pocket money shared? ? Alternately use a fractional approach. 3/8 of the animals are cats. 3/8 of 240 = 90 8 parts = 240 1 part = 30 3 parts = 90 ? 4 parts = £16 1 part = £4 10 parts = £40 2 One quantity given The ratio of boys to girls in a class is 5:6. There are 15 boys. How many girls are there? ? 5 parts = 15 1 part = 3 6 parts = 18

Test Your Understanding 1 [Edexcel Nov 2012] Talil is going to make some concrete mix. He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight.   Talil wants to make 180 kg of concrete mix. Talil has 15 kg of cement 85 kg of sand 100 kg of gravel Does Talil have enough cement, sand and gravel to make the concrete mix? 2 I need to make 300ml of squash, which consists of 1 part concentrate to 5 parts water. How much water do I use? ? 6 parts = 300ml 1 part = 50ml 5 parts = 250ml 3 The ratio of red to blue marbles in a bag is 7:4. If there are 49 red marbles, how many blue marbles are there? 7 parts = 49 1 part = 7 4 parts = 28 ? 4 Frostonia has a population 10,000 more than Yusufland. If the ratio of people in Frostonia to Yusufland is 11:7, what is the population of Yusufland? ? 9 parts = 180kg 1 part (cement) = 20kg 3 parts (sand) = 60kg 5 parts (gravel) = 100kg Insufficient cement. 4 parts = 10,000 1 part = 2,500 7 parts = 17,500 ?

RECAP: Map Scale ? ? Edexcel IGCSE Jan 2014 Method 1: 14cm is 14/4 = 3.5 times bigger. 1km x 3.5 = 3.5km ? Method 2: 4cm : 1km so 1cm : 0.25km So 14cm : 3.5km ? 1km = 1000 x 100 = 100,000cm 100,000 / 4 = 25,000 So ratio is 1 : 25,000

Test Your Understanding 3 16cm on a map represents 12km in real life. a) What does 20cm represent? 15km b) What is the map scale in the form 1:n? 1: 75 000 1 A map scale is 1 : 10 000. Two towns are 14cm apart on the map. What actual distance does this represent in km? 1.4km ? ? ? 2 5cm on a map represents 18km in real life. a) What does 8cm represent? 28.8km b) What is the map scale in the form 1:n? 1: 360 000 4 A map scale is 1 : 50 000. If a supermarket is 3.2km away in real life, how close will it appear on the map? 6.4cm ? ? ?

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 We scale each length by the same amount. i.e.: 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

(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

Harder Problems (both these questions were on past Year 9 landmarks) 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 ?

Harder Problems 3 4 A square is inscribed in a 3-4-5 right-angled triangle as shown. What is the side-length of the square? 5 3 Work out the length of: a) 𝑃𝑄 b) 𝐵𝑃 4 ? 12 7 Suppose the length of the square is 𝑥. Then 3−𝑥 𝑥 = 𝑥 4−𝑥 . Then solve. 𝑃𝑄 12 = 2 3 so 𝑃𝑄=8 𝐵𝐶 3 = 10 12 so 𝐵𝐶=2.5. So 𝐵𝑃=12.5 Many students didn’t realise that the triangle on the left gets flipped upside down on the right, hence got an incorrect answer of 13.6 for 𝐵𝑃. ?

Exercises ? ? ? ? ? 1 2 3 𝑥 5 4 8 5 𝑥 7 𝑥=5.25 𝑟=3.75𝑐𝑚 𝑦=5.6 𝑥=10.8 6 A swimming pool is filled with water. Find 𝑥. 2 3 5𝑐𝑚 5 𝑟 15𝑚 3.75 4 12𝑐𝑚 𝑦 𝑥 1.2𝑚 9𝑐𝑚 3.7𝑚 𝑥 ? 𝑥=5.25 𝑦=5.6 𝑟=3.75𝑐𝑚 ? 𝑥=10.8 ? 1.8𝑚 5 6 4 3 8 5 4 𝑥 𝑥 7 𝑥=1.5 ? 𝑥=4.2 ?

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? A3 𝑦 A4 ? 𝑥 𝑦 = 2𝑦 𝑥 ∴ 𝑥= 2 𝑦 So the length is 2 times greater than the width. A3

Scaling areas and volumes A Savvy-Triangle is enlarged by a scale factor of 3. 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