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One task - and some edifications GyRo-Scop 2010.

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É n, Will Turner - kedvesemmel, a szépséges Liz Swannal - a ka- lózok el ő l menekülvén egy la- katlan szigeten kötöttem ki. A velünk lév ő kincs elrejtésére közösen az alábbiakat találtuk ki …

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I, Will Turner, together with my beautiful sweetheart, Liz Swann, - while escaping from pirates - ended up on a desert island. While trying to hide the treasure we had on us, we made up the following plan...

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After 10 years

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M y belated successor! Please help me by finding the treasure and giving it to those who deserve it. This way you can have (a piece of) it, as well as the pleasure of discovery!

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C A B A’ B’ K

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Making trials? Basic geometry? Rotation of vectors? The scalar product of vectors? Coordinate geometry? Chains of transformations? Lamenting?

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C A B A’ B’ 2 3 4 5 Let’s start from different locations! Idea? Back

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C A B A’ B’ K ß ß’ ß b b c c x x c y y x + y 2 z z Thinking with basic geometry ß + ß’ = 90° The K point is on the perpendicular bisector of the AB, and half way from AB, without respect to the position of C.

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Back

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C A B A’ B’ K b a a’ - b’ a + a’ b – b’ a + b + (a – b)’ 2 a + b 2 (a – b)’ 2 Rotation of vectors a’ – b’ = (a (a – b)’ ? Back a – b

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a b c d c + d 2 1. (a (a – b)(c + d) = ac + ad – bc – bd = ad – bc = 0, because of this c + d a – b 2. (c (c + d)2 d)2 – (a – b)2 b)2 = c2 c2 + 2cd +d2 +d2 – a 2 + 2ab – b2 b2 = 2(cd + ab) = 0, so the length of c + d is equal length of the a – b a – b Scalar product of vectors Back

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A (2a; 0) B (0; 0) Solving the task with the help of coordinate geometry

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C (x; y) A (2a; 0) B (0; 0) A’ (2a + y; 2a – x) B’ (- y; x) (2a – x; - y) (y; 2a – x) K (a; a) Solving the task with the help of coordinate geometry Back

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Chains of transformations P P’ P’’ 90° O1O1 O2O2

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ß ß O A A’ A’’ 1 2 OA = OA’ = OA’’ and AOA’’ = 2 2 + 2ß = 2( + ß)

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Chains of transformations (2nd parth) P P’ P’’ 1 2 3 445° O1O1 O2O2 P*P*

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M y belated successor! You have found it, so the treasure – (part of the) KNOWLEDGE – is yours!

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In general the problems can be solved, but this way they bring new problems up. An Especially Wise Person And here comes the Java (again)!

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While thinking at home study again what you have seen now, make up for the (missing) steps of the with the scalar product, as a voluntary task study the cases of the non-right angled rotation, search for the story of Socrates and the slave, write me your opinion about this lesson including the reasoning as well.

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Thanks for Kristóf Erdélyi Kristóf Erdélyi, pupil of the Árpád Secondary School, the interest, questions and suggestions of dr. Sándor Fridli dr. Sándor Fridli associate professor and dr. István Mezei dr. István Mezei senior lecturer they helped a lot with them in editing the presentation, Ivett Szauftman and Ivett Szauftman, pupil of the Árpád Secondary School. For the patience and attention I am thankful for the audience. I hope that all members of the audience can add their critique so that this presentation can be (even) better. Gyimesi.Robert@arpad.sulinet.hu

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