2GCSE Specification Pack Ref Description 171 Solve problems involving finding lengths in similar shapes.172Understand 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 solids131Convert between units of area137Convert between volume measures, including cubic centimetres and cubic metres
3Similarity 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
4SimilarityThese 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 12Find scale factor: 12 8Then multiply or divide other sides by scale factor as appropriate.𝑥=5× 12 8
5Quickfire 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
6Harder ProblemsWork 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)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?
7(Vote with your diaries) What is the length x? 14x8891012
8(Vote with your diaries) What is the length x? 489x566.5
9(Vote with your diaries) What is the length x? 7.5x1510511.2536.5
10Exercise 172𝑐𝑚1𝐴A swimming pool is filled with water. Find 𝑥.5𝑐𝑚2534𝑟3.7543𝑐𝑚12𝑐𝑚15𝑚𝑦𝑥12𝑐𝑚10𝑐𝑚1.2𝑚9𝑐𝑚3.7𝑚𝐵𝐶?𝑥=5.25𝑦=5.6𝑥?𝑟=3.75𝑐𝑚?𝑩𝑪=𝟖𝒄𝒎 𝑨𝑪=𝟏𝟐.𝟓𝒄𝒎?𝑥=10.8?1.8𝑚5663[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?N1N28545𝑥3𝑥7𝑥=4.2?𝑥=1.5?4[Source: IMO] A square is inscribed in a right-angled triangle as shown. What is the side-length of the square?N3Let 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 𝑏 2 = 1 ℎ 2The 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?𝐴Suppose the length of the square is 𝒙. Then 𝟑−𝒙 𝒙 = 𝒙 𝟒−𝒙 . Solving: 𝒙= 𝟏𝟐 𝟕?By similar triangles 𝑨𝑯= 𝒂𝒉 𝒃Using Pythag on 𝚫𝑨𝑶𝑯:𝒂 𝟐 = 𝒉 𝟐 + 𝒂 𝟐 𝒉 𝟐 𝒃 𝟐Divide by 𝒂 𝟐 𝒉 𝟐 and we’re done.𝐻?𝑎ℎ𝑂𝐵𝑏
11A4/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?A5𝑦A4?𝑥 𝑦 = 2𝑦 𝑥∴ 𝑥= 2 𝑦So the length is 2 times greater than the width.A5
12Scaling areas and volumes A Savvy-Triangle is enlarged by a scale factor of 3 to form a Yusutriangle.2cm?6cm3cm9cm?Area = 3cm2?Area = 27cm2?Length increased by a factor of 3?Area increased by a factor of 9?
13Scaling 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
14Scaling 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?
15B 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
16ExercisesCopy 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?
17Test 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
18Units of Area and Volume We can use the same principle to find how to convert between units of volume and area.1m100cm1m100cm𝑨𝒓𝒆𝒂=𝟏 𝒎 𝟐?𝑨𝒓𝒆𝒂=𝟏𝟎 𝟎𝟎𝟎 𝒎 𝟐?Example:What is 8.3m2 in cm2?8 𝑚 𝑐 𝑚 2× 100 2??
19Quickfire Questions1What is 42cm2 in mm2?42 𝑐𝑚 𝑚 𝑚 25What is 5.1cm2 in mm2?5.1 𝑚 𝑚 𝑚 2× 10 2?× 10 2???What is 2m2 in mm2?2 𝑚 𝑚 𝑚 2What is 2km3 in m3?2 𝑘𝑚 𝑚 326×?×???What is 3m3 in cm3?3 𝑚 𝑚 𝑚 2What is 4.25m2 in mm3?4.25 𝑚 𝑚 𝑚 237× 100 3?×???4What is 13cm3 in mm3?13 𝑚 𝑚 𝑚 28What is 10.01km2 in mm2?10.01 𝑘𝑚 𝑚 𝑚 2×?× 10 3???