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Geometry Similarity and Congruency Skipton Girlsβ High School 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 b (flipping is again allowed) b a a a b? Similarity These two triangles are similar. What is the missing length, and why? 5 7.5 ? 8 12 Thereβs two ways we could solve this: The ratio of the left side and bottom side is the same in both cases, i.e.: 5 π₯ = 8 12 12 Find scale factor: 8 Then multiply or divide other sides by scale factor as appropriate. 12 π₯ =5× 8 Quickfire Examples Given that the shapes are similar, find the missing side (the first 3 can be done in your head). 1 2 12 ? 10 32 ? 24 18 15 15 20 4 3 17 24 11 20 25 ? 40 30 25.88 ? Harder Problems 1 In the diagram BCD is similar to triangle ACE. Work out the length of BD. Work out with your neighbour. 2 The diagram shows a square inside a triangle. DEF is a straight line. What is length EF? (Hint: youβll need to use Pythag at some point) π΅π· 7.5 = 4 10 ?β π΅π· = 3 Since EC = 12cm, by Pythagoras, DC = 9cm. Using similar triangles AEF and CDE: 15 ? πΈπΉ = 9 12 Thus πΈπΉ = 20 Exercise 1 7 1 2 5ππ π 5 π¦ π₯ 12ππ 15π 10ππ 9ππ π΅ π₯ = 5.25 π¦ = 5.6 ? ? 5 6 5 ? π π» π π β π N2 5 3 ? π₯ = 4.5 ππ π By similar triangles π¨π― = Using Pythag on π«π¨πΆπ―: ππ ππ π π π =π + π π Divide by ππ ππ and weβre done. ? β π΅ ? 1.8π π₯ = 10.8 N1 Let π and π be the lengths of the two shorter sides of a right-angled triangle, and let β be the distance from the right angle to the hypotenuse. 1 1 1 Prove 2 + 2 = 2 π π₯ 6 3 7 π₯ = 4.2 π΄ ? ? 4 π₯ πΆ π©πͺ = πππ π¨πͺ = ππ. πππ π₯ N3 1.2π 3.7π π = 3.75ππ 8 A swimming pool is filled with water. Find π₯. 4 3ππ 4 3.75 12ππ 3 2ππ π΄ [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 π to area π? 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? The two unlabelled triangles Suppose the length of the πβπ square is π. Then = π . πβπ ? Solving: π = π ππ π 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. π·: πΈ = π: ππ ? A4/A3/A2 paper A4 A5 π¦ A5 π₯ βAβ 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? π₯ 2π¦ = π¦ π₯ β΄ π₯? = 2π¦ So the length is 2 times greater than the width. GCSE: Congruent Triangles Objective: Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles using formal arguments. What is congruence? These triangles are similar.? They are the same shape. These triangles are congruent. ? They are the same shape and size. (Only rotation and flips allowed) Starter Suppose two triangles have the side lengths. Do the triangles have to be congruent? Yes, because the all the angles are determined by the sides. ? Would the same be true if two quadrilaterals had the same lengths? No. Square and rhombus have same side lengths but are different shapes. ? In pairs, determine whether comparing the following pieces of information would be sufficient to show the triangles are congruent. 3 sides the same. οΌCongruent Two sides the same and angle between them. οΌCongruent ? d c b a All angles the same. ο΄ Not necessarily ? Congruent (but Similar) Two angles the same and Two sides the same and a side the same. angle not between them. οΌCongruent ? ο΄ Not necessarily ? Congruent (weβll see why) Proving congruence GCSE papers will often ask for you to prove that two triangles are congruent. Thereβs 4 different ways in which we could show this: ! a SAS ? Two sides and the included angle. b ASA Two angles and a? side. c SSS Three sides. d RHS ? ? Right-angle, hypotenuse and another side. Proving congruence Why is it not sufficient to show two sides are the same and an angle are the same if the side is not included? Try and draw a triangle with the same side lengths and indicated angle, but that is not congruent to this one. Click to Reveal In general, for βASSβ, there are always 2 possible triangles. What type of proof For triangle, identify if showing the indicating things are equal (to another triangle) are sufficient to prove congruence, and if so, what type of proof we have. This angle is known from the other two. ο» SSS ASA ο» SSS ASA ο» SAS RHS SAS RHS ο» SSS SAS ASA RHS SSS SAS ASA RHS ο» SSS ASA ο» SAS RHS SSS ASA SAS RHS ο» SSS ASA ο» SAS RHS SSS ASA SAS RHS Example Proof Nov 2008 Non Calc STEP 1: Choose your appropriate proof (SSS, SAS, etc.) STEP 2: Justify each of three things. STEP 3: Conclusion, stating the proof you used. Solution: β’ β’ β’ β’ Bro Tip: Always start with 4 bullet points: three for the three letters in your proof, and one for your conclusion. π΄π· = πΆπ· as given π΄π΅ = π΅πΆ as given ? π΅π· is common. β΄ Ξπ΄π·π΅ is congruent to ΞπΆπ·π΅ by SSS. Check Your Understanding π΄ π΅ π΄π΅πΆπ· is a parallelogram. Prove that triangles π΄π΅πΆ and π΄πΆπ· are congruent. (If you finish quickly, try proving another way) πΆ π· Using πππ: β’ β’ β’ β’ Using π΄ππ΄: Using ππ΄π: π΄πΆ is common. π΄π· = π΅πΆ as opposite sides of parallelogram are equal in length. π΄π΅ = π·πΆ for same reason. β΄ Triangles π΄π΅πΆ and π΄πΆπ· are congruent by SSS. ? β’ β’ β’ β’ π΄π· = π΅πΆ as opposite sides of parallelogram are equal in length. β π΄π·πΆ = β π΄π΅πΆ as opposite angles of parallelogram are equal. π΄π΅ = π·πΆ as opposite sides of parallelogram are equal in length. β΄ Triangles π΄π΅πΆ and π΄πΆπ· are congruent by SAS. ? β’ β’ β’ β’ β π΄π·πΆ = β π΄π΅πΆ as opposite angles of parallelogram are equal. π΄π΅ = π·πΆ as opposite sides of parallelogram are equal in length. β π·π΄πΆ = β π΄πΆπ΅ as alternate angles are equal. β΄ Triangles π΄π΅πΆ and π΄πΆπ· are congruent by ASA. ? (if multiple parts, only do (a) for now) NOTE Exercises Q1 ? Exercises Q2 AB = AC (π΄π΅πΆ is equilateral triangle) AD is common. ADC = ADB = 90°. Therefore triangles congruent by RHS. ? Since π΄π·πΆ and π΄π·π΅ are congruent triangles, π΅π· = π·πΆ. π΅πΆ = π΄π΅ as π΄π΅πΆ is equilateral. 1 1 Therefore π΅π· = π΅πΆ = π΄π΅ ? 2 2 Congruent Triangles Q3 ? Exercises Q4 BC = CE equal sides CF = CD equal sides BCF = DCE = 150o BFC is congruent to ECD by SAS. ? So BF=ED (congruent triangles) BF = EG ( opp sides of parallelogram) ? (2) Check Your Understanding What are the four types of congruent triangle proofs? SSS, SAS, ASA (equivalent to AAS) ? and RHS. What should be the structure of our proof? Justification of each of the three letters, followed by ? proof type we used. conclusion in which we state which What kinds of justifications can be used for sides and angles? Circle Theorems, βcommonβ sides, alternate/corresponding angles, properties of parallelograms, sides/angles of regular ? polygon are equal. Using completed proof to justify other sides/angles In this proof, there was no easy way to justify that π΄π΅ = πΆπ·. However, once weβve completed a congruent triangle proof, this provides a justification for other sides and angles being the same. We might write as justification: βAs triangles ABD and DCA are congruent, π΄π΅ = πΆπ·.β Exercises Q2 We earlier showed π΄π·πΆ and π΄π·π΅ are congruent, but couldnβt at that point use π΅π· = π·πΆ because we couldnβt justify it. AB = AC (π΄π΅πΆ is equilateral triangle) AD is common. ADC = ADB = 90°. Therefore triangles congruent by RHS. Since π΄π·πΆ and π΄π·π΅ are congruent triangles, π΅π· = π·πΆ. π΅πΆ = π΄π΅ as π΄π΅πΆ is equilateral. 1 1 Therefore π΅π· = π΅πΆ = π΄π΅ ? 2 2 Exercises Q4 BC = CE equal sides CF = CD equal sides BCF = DCE = 150o BFC is congruent to ECD by SAS. So BF=ED (congruent triangles) BF = EG ( opp sides of parallelogram) ? (2)