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December 2006 - January 2007 Back to Puzzle-of-the-Month page | Home  Difficulty level:   , basic trigonometry knowledge.
Puzzle # 109
Italiano Français Print and cut out the seven right-triangle shapes below and join them together to form a square (no overlappings!). If the hypotenuse of the largest right triangle is 10 units, what is the area of the whole square?  It is not hard to solve this problem using trigonometry. The basic trigonometric functions are functions of an angle; they are important when studying triangles and are commonly defined as ratios of two sides of a right triangle containing the angle. We use the following names for the sides of the right triangle: - The hypotenuse is the longest side of a right triangle. - The opposite side is the side opposite to the angle we are interested in. - The adjacent side is the side that is in contact with the angle we are interested in. Some fundamental trigonometric functions 1) The sine of an angle is the ratio of the opposite side to the hypotenuse: sin(A) = a / h, and h·sin(A) = a 2) The cosine of an angle is the ratio of the adjacent side to the hypotenuse: cos(A) = b / h, and h·cos(A) = b 3) The tangent of an angle is the ratio of the opposite side to the adjacent side: tan(A) = a / b (The sine, cosine and tangent ratios can be remembered by SOH, CAH, TOA -- Sine-Opposite-Hypotenuse, Cosine-Adjacent-Hypotenuse, Tangent-Opposite-Adjacent) Some fundamental trigonometric identities 4) sin(A) / cos(A) = tan(A) 5) sin2(A) + cos2(A) = 1 The right triangles of the puzzle 109 seem to be similar, that is, they all have at least two angles equal: 90° and x. By rotation we can fan the triangles out around their vertex (as shown below), making the adjacent side to the angle x of some triangles merge with the hypothenuse of their adjacent right triangle. Now, using the basic trigonometric functions seen above, we can give a value to each side of the triangles as follows: But the problem is that we do not know the exact value of x. We can only empirically guess that x is approximately /6 and must be 29° < x < 31°, as the drawing suggests: However, with those triangles it is possible to form the following square: The sides of the square must have the same length, thus we obtain the following equation: 10cos(x) = 10cos7(x) + 10sin(x), simplified in 1 = cos6(x) + tan(x) and also: 10cos(x) = 10sin(x)[cos2(x) + cos4(x) + cos6(x)], simplified in 1 = tan(x)[cos2(x) + cos4(x) + cos6(x)] Thanks to the above equations and to the following exact trigonometric constants cos(30) = 3/2 and tan(30) = 3/3, we can already prove that x CANNOT be equal to 30°, in fact: 1 = 3/3[( 3/2)2 + ( 3/2)4 + ( 3/2)6], and then 1 ≠ 3/3[(3/4) + (9/16) + (27/64)] = 111 3/192 (an irrational number multiplied by a rational number can never be equal to an integer!) With some patience and the help of trigonometric tables (or a scientific calculator), we can however approach the real value of x which is ± 29.68. Knowing x it is now easy to find the area of the square: [10cos(29.68)]2 = 75.482 units square To conclude, puzzles don't always have precise solutions or can be systematically solved by defined mathematical processes. This puzzle shows you that the trial-and-error process may also be important to puzzle solving. The winner of the puzzle of the month is: Thierry LEBORDAIS, France. Congratulations Thierry! © 2002 G. Sarcone, www.archimedes-lab.org You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes. You're encouraged to expand and/or improve this article. Send your comments, feedbacks or suggestions to Gianni A. Sarcone.

Previous puzzles of the month...  August 98: the irritating 9-piece puzzle September 98: the impossible squarings October 98: the multi-purpose hexagon November 98: Pythagora's theorem December 98: the cunning areas January 99: less is more (square roots) February 99: another square root problem... March 99: permutation problem... April 99: minimal dissections July 99: jigsaw puzzle August 99: logic? Schmlogic... September 99: hexagon to disc... Oct-Nov 99: curved shapes to square... Dec-Jan 00: rhombus puzzle... February 00: Cheeta tessellating puzzle... March 00: triangular differences... Apr-May 00: 3 smart discs in 1... July 00: Funny tetrahedrons... August 00: Drawned by numbers... September 00: Leonardo's puzzle... Oct-Nov 00: Syntemachion puzzle... Dec-Jan 01: how many squares... February 01: some path problems... March 01: 4D diagonal... April 01: visual proof... May 01: question of reflection... June 01: slice the square cake... July 01: every dog has 3 tails... Aug 01: closed or open... Sept 01: a cup of T... Oct 01: crank calculator... Nov 01: binary art... Dec 01-Jan 02: egyptian architecture... Feb 02: true or false... March 02: enigmatic solids... Apr 02: just numbers... May 02: labyrinthine ways... June 02: rectangle to cross... July-Aug 02: shaved or not... Sept 02: Kangaroo cutting... Oct 02: Improbable solid... Dec-Jan 03: Hands-on geometry Feb-Mar 03: Elementary my dear... Apr-May 03: Granitic thoughts June-July 03: Bagels... September 03: Larger perimeter... Oct-Nov 2003: square vs rectangle Dec-Jan 04: curvilinear shape... February 04: a special box March 04: magic 4 T's... April O4: inscribed rectangle May 04: Pacioli puzzle... June 04: pizza's pitfalls October 04: Odd triangles February 05: Same pieces May-June 05: stairs to square July-August 05: cheese! Sept-Oct 05: magic star Dec-Jan 06: red monad Feb-March 06: cows and chickens Apr-May 06: intriguing probabilities June-July 06: squared strip Aug-Sept 06: precision balance Back to Puzzle-of-the-Month page | Home  MATEMAGICA Blackline masters for making over 25 funny math puzzles! A collection of incredible puzzles and optical illusions with easy-to-follow instructions (in Italian). Ideal for math workshops. More info...
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