Problem 25
Let a,b,c be the three sides of a triangle. They are positive real numbers.
A triangle is possible if and only if all 3 of these conditions are true:
Phrased another way: pick any two sides and add them up. Their sum must be larger than the third side.
In this case we have a = 8, b = 9, c = 11
- a+b = 8+9 = 17 is larger than c = 11. So a+b > c is true
- a+c = 8+11 = 19 is larger than b = 9. This makes a+c > b true
- b+c = 9+11 = 20 is larger than a = 8. So b+c > a is true
We found all three inequalities to be true. Therefore, a triangle is possible due to the triangle inequality theorem.
Answer: A triangle is possible.
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Problem 26
Now we have
Let's use the same idea discussed in problem 25.
- a+b = 7.4+8.1 = 15.5 is larger than c = 9.8; a+b > c is true
- a+c = 7.4+9.8 = 17.2 is larger than b = 8.1; a+c > b is true
- b+c = 8.1+9.8 = 17.9 is larger than a = 7.4; b+c > a is true
Answer: A triangle is possible.
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Bonus Section:
Let's say we had these side lengths
a+b = 1+5 = 6 is NOT larger than c = 10, which means a triangle isn't possible for this example scenario.
I recommend making slips of paper 1 inch, 5 inches and 10 inches long. The 1 inch and 5 inch slips of paper come up short compared to the 10 inch side, which is why a full triangle is not possible here.