Which three lengths CANNOT be the lengths of the sides of a triangle
25m, 16m, 11m
8m, 9m, 10m
7m, 7m, 13m
17m, 4m, 10m
To determine whether three lengths can form the sides of a triangle, we can use the triangle inequality theorem. The theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's apply this theorem to each set of lengths:
1. \( 25m, 16m, 11m \)
For these lengths to form a triangle, the following inequalities must all be true:
- \( 25 < 16 + 11 \) (which simplifies to \( 25 < 27 \), true)
- \( 16 < 25 + 11 \) (which simplifies to \( 16 < 36 \), true)
- \( 11 < 25 + 16 \) (which simplifies to \( 11 < 41 \), true)
Since all three conditions are true, these lengths can form a triangle.
2. \( 8m, 9m, 10m \)
These lengths must meet the following conditions:
- \( 8 < 9 + 10 \) (which simplifies to \( 8 < 19 \), true)
- \( 9 < 8 + 10 \) (which simplifies to \( 9 < 18 \), true)
- \( 10 < 8 + 9 \) (which simplifies to \( 10 < 17 \), true)
All conditions are met, so these lengths also can form a triangle.
3. \( 7m, 7m, 13m \)
For these lengths to form a triangle, the following must be true:
- \( 7 < 7 + 13 \) (which simplifies to \( 7 < 20 \), true)
- \( 7 < 7 + 13 \) (which simplifies to \( 7 < 20 \), true)
- \( 13 < 7 + 7 \) (which simplifies to \( 13 < 14 \), true)
All conditions are met, so these lengths are also able to form a triangle.
4. \( 17m, 4m, 10m \)
To form a triangle, we need:
- \( 17 < 4 + 10 \) (which simplifies to \( 17 < 14 \), false)
- \( 4 < 17 + 10 \) (which simplifies to \( 4 < 27 \), true)
- \( 10 < 17 + 4 \) (which simplifies to \( 10 < 21 \), true)
The first inequality is not true; therefore, \( 17m, 4m, 10m \) cannot form the sides of a triangle because the sum of the two shorter lengths is not greater than the length of the longest side.
The set of lengths that cannot form a triangle is \( 17m, 4m, 10m \).