Presentation on theme: "GCSE Ratio, Scale and Similar Triangles"— Presentation transcript:
1GCSE Ratio, Scale and Similar Triangles Dr J FrostLast modified: 22nd February 2014
2RECAP: Ratio ? ? ? Most ratio problems come in three flavours: 1 Total given.3Difference givenThe 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 = 908 parts = 2401 part = 303 parts = 90?4 parts = £161 part = £410 parts = £402One quantity givenThe ratio of boys to girls in a class is 5:6. There are 15 boys. How many girls are there??5 parts = 151 part = 36 parts = 18
3Test 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 has15 kg of cement85 kg of sand100 kg of gravelDoes Talil have enough cement, sand and gravel to make the concrete mix?2I need to make 300ml of squash, which consists of 1 part concentrate to 5 parts water. How much water do I use??6 parts = 300ml1 part = 50ml5 parts = 250ml3The 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 = 491 part = 74 parts = 28?4Frostonia 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 = 180kg1 part (cement) = 20kg3 parts (sand) = 60kg5 parts (gravel) = 100kgInsufficient cement.4 parts = 10,0001 part = 2,5007 parts = 17,500?
4RECAP: 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 : 0.25kmSo 14cm : 3.5km?1km = 1000 x 100 = 100,000cm100,000 / 4 = 25,000So ratio is 1 : 25,000
5Test Your Understanding 316cm on a map represents 12km in real life.a) What does 20cm represent?15kmb) What is the map scale in the form 1:n?1:1A map scale is 1 : Two towns are 14cm apart on the map. What actual distance does this represent in km?1.4km???25cm on a map represents 18km in real life.a) What does 8cm represent?28.8kmb) What is the map scale in the form 1:n?1:4A map scale is 1 : If a supermarket is 3.2km away in real life, how close will it appear on the map?6.4cm???
6Similarity 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)bbbaaa
7SimilarityThese two triangles are similar. What is the missing length, and why?5?7.5812There’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 12We scale each length by the same amount. i.e.:𝑥 5 = 12 8
8Quickfire ExamplesGiven that the shapes are similar, find the missing side (the first 3 can be done in your head).121012?32?241518152043172411204025?25.88?30
9(Vote with your diaries) What is the length x? 14x8891012
10(Vote with your diaries) What is the length x? 489x566.5
11(Vote with your diaries) What is the length x? 7.5x1510511.2536.5
12Harder 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)1In the diagram BCD is similar to triangle ACE. Work out the length of BD.2Since EC = 12cm, by Pythagoras, DC = 9cm. Using similar triangles AEF and CDE:15 9 = 𝐸𝐹 12Thus 𝐸𝐹=20?𝐵𝐷 4 = → 𝐵𝐷=3?
13Harder Problems34A square is inscribed in a right-angled triangle as shown. What is the side-length of the square?53Work out the length of:a) 𝑃𝑄b) 𝐵𝑃4?12 7Suppose the length of the square is 𝑥. Then 3−𝑥 𝑥 = 𝑥 4−𝑥 . Then solve.𝑃𝑄 12 = so 𝑃𝑄=8𝐵𝐶 3 = so 𝐵𝐶=2.5. So 𝐵𝑃=12.5Many 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.?
14Exercises ? ? ? ? ? 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.235𝑐𝑚5𝑟15𝑚3.75412𝑐𝑚𝑦𝑥1.2𝑚9𝑐𝑚3.7𝑚𝑥?𝑥=5.25𝑦=5.6𝑟=3.75𝑐𝑚?𝑥=10.8?1.8𝑚5643854𝑥𝑥7𝑥=1.5?𝑥=4.2?
15A4/A3/A2 paper𝑥“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?A3𝑦A4?𝑥 𝑦 = 2𝑦 𝑥∴ 𝑥= 2 𝑦So the length is 2 times greater than the width.A3
16Scaling areas and volumes A Savvy-Triangle is enlarged by a scale factor of 3.2cm?6cm3cm9cm?Area = 3cm2?Area = 27cm2?Length increased by a factor of 3?Area increased by a factor of 9?
17Scaling 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 XShape YBro 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?75m2Example: 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 AShape BLength:Area:5cm?×210cm??×430cm2120cm2
18Scaling 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 AShape B?Length:Area:Volume:×3×9?30cm2270cm2×27?80cm32160cm3?
19B 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?BAVolume = 10cm3Volume = 640cm3?Solid ASolid BLength:Area:Volume:×4Answer = 320cm2×1620cm2320cm2×6410cm3640cm3
20ExercisesCopy the table and determine the missing values. 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 𝟑15Shape A Shape BLength:Area:Volume:3cm5cm210cm3×26cm20cm280cm3×4???×8??2Determine the missing values. 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 =𝟗𝟔𝒄𝒎𝟑6Shape A Shape BLength:Area:Volume:5m8m212m3×3?15m72m2324m3?×9?×27???3Determine the missing values.Shape A Shape BLength:Area:Volume:1cm4cm23cm3×5?5cm100cm2375cm3??×25??×125?7The 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 𝒚4Determine the missing values.Shape A Shape BLength:Area:Volume:6m8m210cm3×1.5??9m18m233.75cm3?×2.25??×3.375?
21Test Your Understanding Bro Hint: Scaling mass is the same as scaling what? Volume??Scale factor of area: = 25 9Scale factor of length: = 5 3Scale factor of volume/mass: =500÷ =𝟏𝟎𝟖𝒈