Answer:
- shortest: 11 inches
- longest: 39 inches
Step-by-step explanation:
Given two side lengths for a triangle, the triangle inequality tells you the third side must have a length between the difference and the sum of the given sides.
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A. minimum length
The difference of the given sides 25 and 15 is ...
25 -15 = 10
The smallest integer greater than 10 is 11.
The third side must be at least 11 inches long.
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B. maximum length
The sum of the given sides is ...
25 +15 = 40
The largest integer smaller than 40 is 39.
The third side must be at most 39 inches long.
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Additional comment
Formally, the triangle inequality tells you that sides a, b, c must satisfy ...
a +b > c and b +c > a and c +a > b
When we're considering the length of third side (c), there are 3 cases to consider. 1) c is shortest; 2) c is between the other lengths; 3) c is longest. Without loss of generality, we can assume a > b.
1) c is shortest:
- a+b>c is always true.
- b+c>a ⇒ c>a-b (the positive difference)
- c+a>b ⇒ c>b-a (the negative difference)
conclusion: c > a-b
2) a > c > b:
- a+b>c is always true.
- b+c>a ⇒ c>a-b (the positive difference)
- c+a>b ⇒ c>b-a (the negative difference)
conclusion: c > a-b
3) c is longest:
- a+b>c (the sum)
- b+c>a ⇒ c>a-b (the difference will be less than the sum)
- c+a>b ⇒ c>b-a (the negative difference will be less than the sum)
conclusion: a -b < c < a+b . . . . . . for a>b
