Expert: Michael Tadesse - 5/10/2009

Question
Hi Mr. Tadesse,

I am in Grade 9 and am having trouble with this question:

A triangle can be formed having side lengths 4, 5 and 8. It is impossible, however, to construct a triangle with side lengths 4, 5 and 9. John has 8 sticks, each having an interger length. He observes that he cannot forma  triangle using any three of these sticks as side lengths. The shortest possible length of the longest of the wight sticks is:

a) 20    B) 21   C) 22   D) 23   E)24.

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I understand the concept but I have no idea where to start. Should I try each of the possiblities? How do I go about answering this question?

Answer


  
Hello Susan

I am very interested in your question. I believe it baffled you for some time, but don’t worry I can show you how to solve it.

First grasp this concept. Any triangle has three sides. The sum of any of the two sides must always be larger than the remaining one. This applies to all triangles.



That is    A + B > C, B + C > A,   A + C > B

So Susan, you don't have to try all the possibilities, In fact you could hardly finish checking all, because there are exactly 336 combinations of the eight sticks to be grouped into three.

But try to assume that the shortest length the longest of the eight sticks would have is the sum of the nearest long sticks. In that way we can succeed finding the length because; if it is (the length of the longest stick) equal or greater than the sum the length of any two sticks we can never construct a triangle. Start with the simple number. Let the length of the first two sticks be 1.
                                          1 + 1 = 2 (the length of the third stick for the three sticks not to form a triangle)
                                          2 + 1 = 3 (the length of the fourth stick, because 2 + 1 can’t exceed 3)
                                          3 + 2 = 5 (the fifth stick must have this length)
                                          5 + 3 = 8 (stick 6)
                                          8 + 5 = 13(stick 7)
                                         13+ 8 = 21(the last stick must have a length of 21 in order for any of the 8 sticks not form a triangle)

Try to add any two numbers and find a sum larger than 21 in this combination. You never find one.

So the longest of the eight sticks must be 21 Units long.


If you can’t understand it please let me know I can help. Don’t hesitate to ask me more I can help you learn how to solve another problems.

Cordially,
Michael Tadesse.