Triangle Inequality TheoremDate: 03/09/2001 at 16:30:56 From: Cprime Realista Subject: Non-isosceles triangle The lengths of the sides of a non-isosceles triangle, in size order, are 5, x, and 15. What are all possible integral values of x? My guess would be that the number is somewhere between 6 and 14. Date: 03/13/2001 at 09:58:13 From: Doctor Patti Subject: Re: Non-isosceles triangle Hello Cprime, Thank you for writing to Dr. Math. There is a theorem in geometry (the triangle inequality theorem) that states: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's take a look at an "open" triangle with sides x, y, and z / x / \ / \ y / \ ------------------------ z To see the reasoning behind the theorem, think of sides x and y 'folding' down toward side z. If the sum of the lengths of x and y is not greater than the length of z, they will lie flat on z, and this would not make a triangle. So, from the theorem x + y > z y + z > x x + z > y Let's look at a triangle with sides 2, 8, and x. What are all possible integral values of x? x would need to be greater than 6; otherwise sides with 2 and 6 would lie flat on the side with length 8. x would also need to be less than 10; otherwise sides 2 and 8 would lie flat on the side with length 10. Therefore, 6 < x < 10 or x is between, but does not include, 6 and 10 When working with this type of problem I think to myself: the length of x, the third side, needs to be LESS than than the sum of the two sides and GREATER than the difference of the two sides. I hope this helps. Please write back if you have any other questions. - Doctor Patti, The Math Forum http://mathforum.org/dr.math/ |
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