One task - and some edifications

GyRo-Scop2010.

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 ...

After 10 years

My 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!

C

A

B

A’

B’

K

Making trials?Basic geometry?

Rotation of vectors?The scalar product of vectors?

Coordinate geometry?Chains of transformations?

Lamenting?

C

A

B

A’

B’

2

3

45

Let’s start from different locations!

Idea?

Back

C

A

B

A’

B’

K

ß

ß’

ß

ß’b

b

c

c

x

x

cy

y

x + y 2

zz

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.

Back

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 – b)’ ?

Back

a – b

a

b

c

d

c + d 2

1. (a – b)(c + d) = ac + ad – bc – bd = ad – bc = 0, because of this c + d a – b

2. (c + d)2 – (a – b)2 = c2 + 2cd +d2 – a2+ 2ab – 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

A (2a; 0)

B (0; 0)

Solving the task with the help of coordinate geometry

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

Chains of transformations

P

P’

P’’

90°

90°

O1

O2

ßß

OA

A’

A’’

1

2

OA = OA’ = OA’’and

AOA’’ = 2 + 2ß = 2( + ß)

Chains of transformations (2nd parth)

P

P’

P’’

1

2 3

4 45°

45°

O1

O2

P*

My belated successor! You have found it, so the treasure –

(part of the)KNOWLEDGE –

is yours!

In general the problems can be solved,but this way they bring new problems up.

An Especially Wise Person

And here comes And here comes the the Java Java (again)!(again)!

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.

ThanksThanksfor

Kristóf ErdélyiKristóf Erdélyi, pupil of the Árpád Secondary School,the interest, questions and suggestions ofdr.dr. Sándor Fridli Sándor Fridli associate professor and

dr. István Mezei dr. István Mezei senior lecturerthey helped a lot with them in editing the

presentation,and Ivett Szauftman Ivett Szauftman,

pupil of the Árpád Secondary School.

For the patience and attention I amtthankfulhankful

for the audience.

I hopeI hopethat all members of the audience can add their critique so that this presentation can be (even)

better.

Gyimesi.Robert@arpad.sulinet.hu