Question 457725: How can I know whether a triangle is acute, obtuse, or right, based on three side lengths?
Example, 6 cm, 8 cm, 9 cm.
I know it is not a right triangle, because the pythagorean theorem does not work.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You are on the right track.
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Take the two short sides of the triangle and use them in the Pythagorean theorem. If you did that, you found that in a right triangle with legs 6 and 8, the hypotenuse would be 10 because 6^2 = 36 and 8^2 = 64 and the sum of 36 and 64 is 100. The hypotenuse is then the square root of 100 or 10.
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But the hypotenuse 10 is longer than the longest side you were given ... 9. If you sketch the 6 - 8- 9 triangle you will find that because the longest side is shorter than 10, the triangle is acute. In order for a triangle to be obtuse, the longest side must be bigger than the hypotenuse that would make the two short sides a right triangle.
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So in recipe form for problems such as this:
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1) Take the two short sides of the given triangle and use them to calculate the hypotenuse of a right triangle.
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2) Compare the hypotenuse to the longest side given in the problem. If the longest given side is shorter than the hypotenuse, the given triangle is acute. If it is longer than the hypotenuse, the given triangle is obtuse.
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A word of caution. Always be sure you have a triangle. In order for there to be a triangle, the sum of the two short sides must be longer than the long side. For example: If I said a triangle had sides of 2,7, and 11 you could immediately say I was wrong because you know that 2+7 is less than 11 so you cannot make a triangle with those dimensions. You can try this by cutting three strips of paper, one 2 inches long, one 7 inches long, and one 11 inches long. When you try to form a triangle by using these three strips as sides, you will see why this cannot be done.
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For the problem you were given, you could make the triangle obtuse by lengthening the longest side. With sides of 6 and 8 you know the hypotenuse is 10 as we discussed above. Therefore, you could make the triangle obtuse by making the longest side be longer than 10. But now you know that the longest side cannot be 14 or more. Why? Because 6 + 8 equals 14 and the longest side must be shorter than the sum of the two shortest sides. Therefore, with sides of 6 and 8, the triangle will be obtuse if the longest side is greater than 10 but less than 14.
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Probably more than you wanted to know about triangles, but hope this helps you understand triangles a little more.
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