Question 124665: My question goes like this..."With two sticks of length 6 cm. and 9cm., how many triangles can be formed with the two sticks and a third stick with integral length?"
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! The answer to this is based upon the fact that the sum of the two short sides of a triangle must
be greater than the long side.
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You are given two sides, of which 9 is the longer. Therefore, the third side must be greater
than 3. Why? because 6 + 3 just equals 9. If you laid the side 6 in a direct line with the side
3, they would form a line 9 units long which would just be congruent with the third side 9.
To form a triangle, the two short sides must together form a line that is longer than 9.
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But what happens if 9 is not the long side of the triangle? What happens if the unknown third
stick is the long side? This unknown long side must be shorter than 15, because the other two
sides (9 and 6) must add up to be longer than the third side. So, in integral values, the
third stick must be 14 or shorter.
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So the values of the third stick may be anywhere in the interval from 4 to 14 ... including
the values 4 and 14. If you count all the integers in this interval you will find that the
unknown stick can have the values 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14. So 11 triangles
can be formed.
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Hope this helps you to understand the problem.
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