Answer :
To determine the possible lengths of the shortest straw that would form an obtuse triangle with the other two straws of lengths 9 inches and 13 inches, we need to follow these steps:
1. Understand the conditions for forming a triangle:
For three lengths to form a valid triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
2. Conditions for a valid triangle:
- [tex]\( length + 9 > 13 \)[/tex]
- [tex]\( length + 13 > 9 \)[/tex]
- [tex]\( 9 + 13 > length \)[/tex]
3. Conditions for an obtuse triangle:
For a triangle to be obtuse, the square of the longest side must be greater than the sum of the squares of the other two sides. If we denote the three sides as [tex]\(a \leq b \leq c\)[/tex]:
[tex]\[ a^2 + b^2 < c^2 \][/tex]
Let's evaluate this for each given possible length of the shortest straw, where the third length is:
- For a length of 5 inches:
- Check if it forms a valid triangle:
[tex]\[5 + 9 > 13 \rightarrow 14 > 13 \quad (true)\][/tex]
[tex]\[5 + 13 > 9 \rightarrow 18 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 5 \rightarrow 22 > 5 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 5^2 + 9^2 < 13^2 \rightarrow 25 + 81 < 169 \rightarrow 106 < 169 \quad (true) \][/tex]
- For a length of 6 inches:
- Check if it forms a valid triangle:
[tex]\[6 + 9 > 13 \rightarrow 15 > 13 \quad (true)\][/tex]
[tex]\[6 + 13 > 9 \rightarrow 19 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 6 \rightarrow 22 > 6 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 6^2 + 9^2 < 13^2 \rightarrow 36 + 81 < 169 \rightarrow 117 < 169 \quad (true) \][/tex]
- For a length of 7 inches:
- Check if it forms a valid triangle:
[tex]\[7 + 9 > 13 \rightarrow 16 > 13 \quad (true)\][/tex]
[tex]\[7 + 13 > 9 \rightarrow 20 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 7 \rightarrow 22 > 7 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 7^2 + 9^2 < 13^2 \rightarrow 49 + 81 < 169 \rightarrow 130 < 169 \quad (true) \][/tex]
- For a length of 8 inches:
- Check if it forms a valid triangle:
[tex]\[8 + 9 > 13 \rightarrow 17 > 13 \quad (true)\][/tex]
[tex]\[8 + 13 > 9 \rightarrow 21 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 8 \rightarrow 22 > 8 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 8^2 + 9^2 < 13^2 \rightarrow 64 + 81 < 169 \rightarrow 145 < 169 \quad (true) \][/tex]
- For a length of 9 inches:
- Check if it forms a valid triangle:
[tex]\[9 + 9 > 13 \rightarrow 18 > 13 \quad (true)\][/tex]
[tex]\[9 + 13 > 9 \rightarrow 22 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 9 \rightarrow 22 > 9 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 9^2 + 9^2 < 13^2 \rightarrow 81 + 81 < 169 \rightarrow 162 < 169 \quad (true) \][/tex]
Therefore, the lengths of the shortest straw that could form an obtuse triangle with the other two straws are:
- 5 inches
- 6 inches
- 7 inches
- 8 inches
- 9 inches
1. Understand the conditions for forming a triangle:
For three lengths to form a valid triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
2. Conditions for a valid triangle:
- [tex]\( length + 9 > 13 \)[/tex]
- [tex]\( length + 13 > 9 \)[/tex]
- [tex]\( 9 + 13 > length \)[/tex]
3. Conditions for an obtuse triangle:
For a triangle to be obtuse, the square of the longest side must be greater than the sum of the squares of the other two sides. If we denote the three sides as [tex]\(a \leq b \leq c\)[/tex]:
[tex]\[ a^2 + b^2 < c^2 \][/tex]
Let's evaluate this for each given possible length of the shortest straw, where the third length is:
- For a length of 5 inches:
- Check if it forms a valid triangle:
[tex]\[5 + 9 > 13 \rightarrow 14 > 13 \quad (true)\][/tex]
[tex]\[5 + 13 > 9 \rightarrow 18 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 5 \rightarrow 22 > 5 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 5^2 + 9^2 < 13^2 \rightarrow 25 + 81 < 169 \rightarrow 106 < 169 \quad (true) \][/tex]
- For a length of 6 inches:
- Check if it forms a valid triangle:
[tex]\[6 + 9 > 13 \rightarrow 15 > 13 \quad (true)\][/tex]
[tex]\[6 + 13 > 9 \rightarrow 19 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 6 \rightarrow 22 > 6 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 6^2 + 9^2 < 13^2 \rightarrow 36 + 81 < 169 \rightarrow 117 < 169 \quad (true) \][/tex]
- For a length of 7 inches:
- Check if it forms a valid triangle:
[tex]\[7 + 9 > 13 \rightarrow 16 > 13 \quad (true)\][/tex]
[tex]\[7 + 13 > 9 \rightarrow 20 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 7 \rightarrow 22 > 7 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 7^2 + 9^2 < 13^2 \rightarrow 49 + 81 < 169 \rightarrow 130 < 169 \quad (true) \][/tex]
- For a length of 8 inches:
- Check if it forms a valid triangle:
[tex]\[8 + 9 > 13 \rightarrow 17 > 13 \quad (true)\][/tex]
[tex]\[8 + 13 > 9 \rightarrow 21 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 8 \rightarrow 22 > 8 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 8^2 + 9^2 < 13^2 \rightarrow 64 + 81 < 169 \rightarrow 145 < 169 \quad (true) \][/tex]
- For a length of 9 inches:
- Check if it forms a valid triangle:
[tex]\[9 + 9 > 13 \rightarrow 18 > 13 \quad (true)\][/tex]
[tex]\[9 + 13 > 9 \rightarrow 22 > 9 \quad (true)\][/tex]
[tex]\[9 + 13 > 9 \rightarrow 22 > 9 \quad (true)\][/tex]
- If valid, check if the triangle is obtuse:
[tex]\[ 9^2 + 9^2 < 13^2 \rightarrow 81 + 81 < 169 \rightarrow 162 < 169 \quad (true) \][/tex]
Therefore, the lengths of the shortest straw that could form an obtuse triangle with the other two straws are:
- 5 inches
- 6 inches
- 7 inches
- 8 inches
- 9 inches
