AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC) AIMING HIGH GOLDEN PENTAGON In each of these diagrams of a regular pentagon find the ratio of the length shown in red to the length in blue in terms of the Golden Ratio = ($√) Find all the angles in the diagram. Let AE = 1 unit and BE = x units. Which triangles are isosceles? Which triangles are similar? Use similar triangles to give an equation for x and solve the equation. Prove () *) = Prove () (+ = () (, = Prove = Prove =∅
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AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC)
AIMING HIGH
GOLDEN PENTAGON IneachofthesediagramsofaregularpentagonfindtheratioofthelengthshowninredtothelengthinblueintermsoftheGoldenRatio𝜙 = 𝟏
HELP Youmayfindithelpfultomarkalltheanglesof36oinonecolour,anglesof54oinanothercolourandanglesof72oinathirdcolour.Thiswillhelpyoutoseewhichtrianglesareisosceles,andalsotoseewhichpairsoftrianglesaresimilar.Writetheratiosintermsofxand1+xandsimplifytheexpressionstogivequadraticequations.Forsomeoftheseproofsyouwillneedtomanipulatesurds.
NEXT Usingthesamediagramwritedownthevalueofcos36ointermsof√5.
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NOTES FOR TEACHERS SOLUTION Theanglesofaregularpentagonare108°soalltheanglescanbefound.
Hence x2 - x - 1 = 0 and thisquadraticequationhassolutions
Why do this activity? Thisactivityoffersablendofalgebraandgeometrysuitablefor14yearoldsandolderstudents.Thereisscaffoldingtoguidestudentstosolvetheproblemusingthesolutionstoaquadraticequation.Learning objectives Indoingthisactivitystudentswillhaveanopportunityto:• reviewanddeepentheirknowledgeandunderstandingofsimilartriangles;• reviewanddeepentheirknowledgeandunderstandingofratios;• reviewanddeepentheirknowledgeandunderstandingofhowtosolveaquadratic