Geometry Tasks

Arthur Lee, Nov 24, 2014

various tasks sharing with teachers

Table of Contents

  1. Langley Problem
    1. Langley Problem
    2. octadecagon
  2. squares and parallelogram
    1. squares around parallelogram
    2. more squares and parallelograms
    3. pentagonal tiling from squares and parallelograms
  3. pentagonal tiles
    1. Cairo Tiling 1
    2. Cairo Tiling 2
  4. opposite sides and angles of hexagon
    1. opposite sides and angles of hexagon
    2. opposite angles of hexagon
    3. opposite side length of hexagon
    4. parallel opposite sides of hexagon
  5. treasure island
    1. Treasure Island Puzzle
  6. Sangaku
    1. A Sangaku Problem
    2. Sangaku (算額)：Five Circles in a Square (Solution)

1.

Langley Problem

[color=#c51414]Open the task sheet:[/color] https://www.evernote.com/l/AAOQDE63bJFHM6VfNYnAAEnDb-HzWP6sJjk More info about this problem can be found in these slides: https://dl.dropboxusercontent.com/u/4210789/me/mini/Langley/index.html

  • 1. Langley Problem
  • 2. octadecagon

1.1.

Langley Problem

Adjust the blue angles at A and B. Which combinations make it easy / difficult to solve for angle D?
This is a famous problem which looks simple but is in fact extremely difficult to solve with angle chasing. You may have a try, finding the unknown angle D in the figure. [br][br]The task here is not about directly solving this problem. You may consider varying the blue angles to other multiples of 10 degrees. [br][br]When would it be easy or difficult for finding the angle D? [br]When doing so, can you get a sense of what makes this problem difficult or easy to solve?[br][br]You may use the applet to quickly adjust the blue angles and see how the figure changes.[br]Do you have other suggestions of modifying this problem?[br][br]More info about this problem can be found in these slides: [br][url=https://dl.dropboxusercontent.com/u/4210789/me/mini/Langley/index.html]https://dl.dropboxusercontent.com/u/4210789/me/mini/Langley/index.html[/url]
– Arthur Lee

1.2.

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2.

squares and parallelogram

[color=#c51414]Open the task sheet:[/color] https://www.evernote.com/l/AAPkK3E9enNHAKc6KqQPOUW77qJNNl5Hz7I

  • 1. squares around parallelogram
  • 2. more squares and parallelograms
  • 3. pentagonal tiling from squares and parallelograms

2.1.

squares around parallelogram

[color=#c51414]Open the task sheet:[/color][br][url=https://www.evernote.com/l/AAPkK3E9enNHAKc6KqQPOUW77qJNNl5Hz7I]https://www.evernote.com/l/AAPkK3E9enNHAKc6KqQPOUW77qJNNl5Hz7I[/url]
Thebault's Theorem[br][url=http://nrich.maths.org/1939]http://nrich.maths.org/1939[/url]
– Arthur Lee

2.2.

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2.3.

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3.

pentagonal tiles

[color=#c51414]Open the task sheet:[/color] https://www.evernote.com/l/AAO-URTJbMVF5rsObg8eNmgNIEWaahXb5aM More about tiling: https://dl.dropboxusercontent.com/u/4210789/me/mini/tiling/index.html

  • 1. Cairo Tiling 1
  • 2. Cairo Tiling 2

3.1.

Cairo Tiling 1

[color=#c51414]Open the task sheet:[/color][br][url=https://www.evernote.com/l/AAO-URTJbMVF5rsObg8eNmgNIEWaahXb5aM]https://www.evernote.com/l/AAO-URTJbMVF5rsObg8eNmgNIEWaahXb5aM[/url]
– Arthur Lee

3.2.

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4.

opposite sides and angles of hexagon

[color=#c51414]Open the task sheet:[/color] https://www.evernote.com/l/AANyIWZaqvBBaIUzGt7b_T6DZZjW3Og9zOA more info: https://dl.dropboxusercontent.com/u/4210789/me/mini/hexagons/index.html

  • 1. opposite sides and angles of hexagon
  • 2. opposite angles of hexagon
  • 3. opposite side length of hexagon
  • 4. parallel opposite sides of hexagon

4.1.

opposite sides and angles of hexagon

[color=#c51414]Open the task sheet:[/color][br]https://www.evernote.com/l/AANyIWZaqvBBaIUzGt7b_T6DZZjW3Og9zOA
more info:[br]https://dl.dropboxusercontent.com/u/4210789/me/mini/hexagons/index.html
– Arthur Lee

4.2.

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4.3.

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4.4.

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5.

treasure island

[color=#c51414]Open the task sheet:[/color] https://www.evernote.com/l/AAPZAYsI93FDyqPqbOfRDHlw6FtHThNJY4I Find the treasure with a missing gallows. These are links related to this problem: https://www.ma.utexas.edu/users/gilbert/M333L/BinderDay2.pdf http://frink.machighway.com/~dynamicm/hunt.pdf http://jwilson.coe.uga.edu/EMAT6680Su07/McKee/6690/islandtreasure/it1.html http://books.google.com.hk/books?id=et6WMlkQlFcC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false https://www.academia.edu/5600848/Back_to_Treasure_Island

  • 1. Treasure Island Puzzle

5.1.

Treasure Island Puzzle

[color=#c51414]Open the task sheet:[/color][br][url=https://www.evernote.com/l/AAPZAYsI93FDyqPqbOfRDHlw6FtHThNJY4I]https://www.evernote.com/l/AAPZAYsI93FDyqPqbOfRDHlw6FtHThNJY4I[/url]
Find the treasure with a missing gallows.[br]These are links related to this problem:[br][list][br][*]https://www.ma.utexas.edu/users/gilbert/M333L/BinderDay2.pdf[br][br][/*][*]http://frink.machighway.com/~dynamicm/hunt.pdf[br][br][/*][*]http://jwilson.coe.uga.edu/EMAT6680Su07/McKee/6690/islandtreasure/it1.html[br][br][/*][*]http://books.google.com.hk/books?id=et6WMlkQlFcC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false[br][br][/*][*]https://www.academia.edu/5600848/Back_to_Treasure_Island[br][/*][/list]
– Arthur Lee

6.

Sangaku

http://www.cut-the-knot.org/pythagoras/Sangaku.shtml

  • 1. A Sangaku Problem
  • 2. Sangaku (算額)：Five Circles in a Square (Solution)

6.1.

A Sangaku Problem

A Sangaku Problem
A Sangaku Problem
How to obtain the analytic solution by origami?
– Arthur Lee, spkwan

6.2.

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