Question 511350: Which of the following cannot be possible to construct a triangle with the given side lengths?
Answer
a. 28, 34, 39
b. 3, 6, 9
c. 40, 50, 60
d. 35, 120, 125
e. 6, 7, 11
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! The way to do this problem is to use the following 4 steps:
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(1) Look at the possible answers one at a time.
(2) In each answer identify the longest side.
(3) When you have selected the longest side, add the two remaining sides together.
(4) It the sum of the two remaining sides is greater than the longest side, then it is possible to form a triangle.
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Let's work on each answer using the above process steps (2) through (4)
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answer a. Sides are 28, 34, 39. Step (2) tells you to identify the longest side. It is 39. Next, Step (3) tells you to add the two remaining sides. The two smaller sides are 28 and 34. Add them together and you will get 62. Then Step (4) says if the sum of these two remaining sides (in this case the sum is 62) is greater than the longest side (39), then a triangle can be formed. Obviously 62 is greater than 39, so a triangle can be formed using these three sides.
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Skip answer b for the time being.
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answer c. The sides are 40, 50, and 60. The longest side is 60. The sum of the two remaining sides is 40 + 50 = 90. Note that 90 is greater than 60, so a triangle can be formed.
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answer d. Sides of 35, 120, and 125. Longest side = 125. Sum of the two remaining sides is 35 + 120 = 155. The sum 155 is greater than 125 so a triangle can be formed.
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answer e. Sides 6, 7, 11. Longest side = 11. Sum of two other sides is 6 + 7 = 13. 13 is greater than 11, so a triangle can be formed.
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Return to answer b. Sides 3, 6, 9. Longest side = 9. Sum of two other sides is 3 + 6 = 9. So the sum of the two shorter sides (sum = 9) is equal to, but NOT GREATER THAN the longest side. Therefore, a triangle can NOT be formed.
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The answer to this problem is answer b.
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Two other cases should be looked at for the general rule. If there are TWO 'LONGEST' sides (for example an isosceles situation with equal sides 20, 20, and a third shorter side of 5), a triangle can always be formed. You can verify this by selecting one of the longer sides (20) and then see that the sum of the two remaining sides (20 and 5) is greater than the one longer side, so an isosceles triangle can be formed.
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However, if there is one longest side, and the sum of the two equal (and shorter) sides is NOT greater than the longest side, then a triangle CANNOT be formed. For example, the sides are 5, 5, and 15. The longest side is 15 and the sum of the two remaining sides is 10. Since 10 is smaller than 15, an isosceles triangle cannot be formed.
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The same can be said about an equilateral triangle (all sides equal in length). A triangle can always be formed because if you select any one of the sides to be the 'longest', the sum of the two remaining sides is always greater than than the longest side.
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Hope this helps you to understand this problem. It might also help if you used a ruler to sketch out a couple of problems. For example, use a ruler to make a horizontal line 4 inches long in the middle of a sheet of paper. The assume that two other sides are 2 inches and 1 inch in length. Connect one of them to one end of the 4 inch line and the other to the other end of the 4 inch line. It will fairly easy to see why these short sides cannot be rotated such that they will join together to form a triangle with the 4 inch line.
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