Question 1205315
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With the lengths of 10 and 5 with a 55-degree angle between them; 
how many quadrilaterals can be made with such information?
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<pre>
We are given 4 (four) sticks. Two of them are 5 units long; two others are 10 units long.
How many different quadrilateral shapes can be made using these 4 sticks, if the angle 
between each pair (5,10) at vertex is 55 degrees ?


Let "a" denotes a stick of 5 units long, and let "b" denotes a stick of 10 units long.

If the sticks are going in this order "abab", then the quiadrilateral is a PARALLELOGRAM,
since if in a quadrilateral opposite sides are equal, it implies that this quadrilateral
is NECESSARY a parallelogram.


But then there is an angle of 180-55 = 125 degrees between some two adjacent sticks "ab",
so this case is not what we are seeking for.


It implies, that the only allowed order of the sides, which may work, is "aabb".


Such shape does exist, and it is called a "kite".


But a convex kite shape does not work, since then the angle between sides "a" and "b" is greater than 55 degrees.

Hence, the only possible shape, which may work, is a non-convex kite shape "aabb", which has  
angles of 55 degrees between the sides "a" and "b".


I will not go forward with more detailed explanations - I just said enough in order for
a normal person with geometric imagination could restore this figure in his/(her) mind.


Just imagine an obtuse triangle with sides 5 and 10 units long and with the angle 55 degrees between them.

Its third side "c" has the unique length by the cosine law.


Now imagine TWO such triangles, attached along this common side "c" in a way, that they form 
a non-convex kite - and it will give you the required quadrilateral.


          +----------------------------------+
          |    Its shape is an "arrowhead".  |
          +----------------------------------+


So, considered to congruency, only ONE such quadrilateral exists and is possible.    <U>ANSWER</U>
</pre>

Solved.



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Really nice problem. It is non-trivial, has unexpected solution and 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;trains a logic and a geometric imagination, both at the same time.



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