Which sets of potential side lengths make a triangle?

Understand the Problem

The question is asking to determine which sets of potential side lengths can form a triangle based on the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.

Answer

- Makes a Triangle: 2.4 mm, 5.7 mm, 7.9 mm; 3.1 cm, 3.1 cm, 6.1 cm; 18.23 ft, 18.23 ft, 18.23 ft - Does Not Make a Triangle: 3 in, 5 in, 8 in; 5 cm, 7 cm, 13 cm

Steps to Solve

Identify the side lengths

We have the following sets of side lengths to analyze:
- Set 1: $3 , \text{in}, 5 , \text{in}, 8 , \text{in}$
- Set 2: $5 , \text{cm}, 7 , \text{cm}, 13 , \text{cm}$
- Set 3: $2.4 , \text{mm}, 5.7 , \text{mm}, 7.9 , \text{mm}$
- Set 4: $3.1 , \text{cm}, 3.1 , \text{cm}, 6.1 , \text{cm}$
- Set 5: $18.23 , \text{ft}, 18.23 , \text{ft}, 18.23 , \text{ft}$
Apply the triangle inequality theorem

According to the triangle inequality theorem, for any three side lengths (a), (b), and (c):

$$ a + b > c $$ $$ a + c > b $$ $$ b + c > a $$

We will examine each set of lengths to see if they satisfy these conditions.
Evaluate each set
- Set 1: $3 , \text{in}, 5 , \text{in}, 8 , \text{in}$
  - Check:
    - $3 + 5 = 8 \not> 8$ (fails).
- Set 2: $5 , \text{cm}, 7 , \text{cm}, 13 , \text{cm}$
  - Check:
    - $5 + 7 = 12 < 13$ (fails).
- Set 3: $2.4 , \text{mm}, 5.7 , \text{mm}, 7.9 , \text{mm}$
  - Check:
    - $2.4 + 5.7 = 8.1 > 7.9$ (passes).
    - $2.4 + 7.9 = 10.3 > 5.7$ (passes).
    - $5.7 + 7.9 = 13.6 > 2.4$ (passes).
  - Therefore, this set makes a triangle.
- Set 4: $3.1 , \text{cm}, 3.1 , \text{cm}, 6.1 , \text{cm}$
  - Check:
    - $3.1 + 3.1 = 6.2 > 6.1$ (passes).
    - $3.1 + 6.1 = 9.2 > 3.1$ (passes).
    - $3.1 + 6.1 = 9.2 > 3.1$ (passes).
  - Therefore, this set makes a triangle.
- Set 5: $18.23 , \text{ft}, 18.23 , \text{ft}, 18.23 , \text{ft}$
  - All sides are equal, so by property of equilateral triangles:
    - Each condition is satisfied, thus it makes a triangle.
Conclusion

From the analysis:
- Set 3, Set 4, and Set 5 make a triangle.
- Set 1 and Set 2 do not make a triangle.

Makes a Triangle: 2.4 mm, 5.7 mm, 7.9 mm; 3.1 cm, 3.1 cm, 6.1 cm; 18.23 ft, 18.23 ft, 18.23 ft
Does Not Make a Triangle: 3 in, 5 in, 8 in; 5 cm, 7 cm, 13 cm

More Information

The triangle inequality theorem is crucial in determining whether three lengths can form a triangle. Sets of lengths where the sum of any two sides is greater than the third side will always be able to form a triangle.

Tips

Misapplying the inequalities: Remember that the sum must be strictly greater than the third side; equality does not count.
Failing to evaluate all three inequalities: Always check all combinations for completeness.

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