Probability - triangles

We have five lines with lengths of 3 cm, 5 cm, 7 cm, 9 cm, and 11 cm. What is the probability that we will be able to construct a triangle with randomly selected three?

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Dr. Math

For clarity, we list all possible triples and check whether they satisfy the triangle inequality:

1. 3, 5, 7
- 3 + 5 > 7 → 8 > 7 (yes)
- 3 + 7 > 5 → 10 > 5 (yes)
- 5 + 7 > 3 → 12 > 3 (yes)
A triangle can be constructed.

2. 3, 5, 9
- 3 + 5 > 9 → 8 > 9 (no)
A triangle cannot be constructed.

3. 3, 5, 11
- 3 + 5 > 11 → 8 > 11 (no)
A triangle cannot be constructed.

4. 3, 7, 9
- 3 + 7 > 9 → 10 > 9 (yes)
- 3 + 9 > 7 → 12 > 7 (yes)
- 7 + 9 > 3 → 16 > 3 (yes)
A triangle can be constructed.

5. 3, 7, 11
- 3 + 7 > 11 → 10 > 11 (no)
A triangle cannot be constructed.

6. 3, 9, 11
- 3 + 9 > 11 → 12 > 11 (yes)
- 3 + 11 > 9 → 14 > 9 (yes)
- 9 + 11 > 3 → 20 > 3 (yes)
A triangle can be constructed.

7. 5, 7, 9
- 5 + 7 > 9 → 12 > 9 (yes)
- 5 + 9 > 7 → 14 > 7 (yes)
- 7 + 9 > 5 → 16 > 5 (yes)
A triangle can be constructed.

8. 5, 7, 11
- 5 + 7 > 11 → 12 > 11 (yes)
- 5 + 11 > 7 → 16 > 7 (yes)
- 7 + 11 > 5 → 18 > 5 (yes)
A triangle can be constructed.

9. 5, 9, 11
- 5 + 9 > 11 → 14 > 11 (yes)
- 5 + 11 > 9 → 16 > 9 (yes)
- 9 + 11 > 5 → 20 > 5 (yes)
A triangle can be constructed.

10. 7, 9, 11
- 7 + 9 > 11 → 16 > 11 (yes)
- 7 + 11 > 9 → 18 > 9 (yes)
- 9 + 11 > 7 → 20 > 7 (yes)
A triangle can be constructed.

Of the 10 possible triples, 7 satisfy the triangle inequality, and therefore a triangle can be constructed from them.

The probability P is therefore:

P = Number of favorable triplesTotal number of triples = 710 = 0{.}7

The probability that a triangle can be constructed by randomly selecting a triple of line segments is 710 or 70%.

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