To determine the possible lengths of the shortest straw that would form an obtuse triangle with the other two straws, we must use the properties of an obtuse triangle.
An obtuse triangle is one in which one of the angles is greater than 90 degrees. For that to happen, the square of the length of the longest side must be greater than the sum of the squares of the other two sides.
Given:
- The lengths of the two known sides of the triangle are 12 inches and 13 inches.
Let the unknown length of the shortest straw be denoted by [tex]\( x \)[/tex].
Given possible lengths for [tex]\( x \)[/tex]:
- 5 inches
- 6 inches
- 7 inches
- 8 inches
- 9 inches
We need to check if each length can form an obtuse triangle with the given sides 12 inches and 13 inches. To do this, we'll use the condition [tex]\( a^2 + b^2 < c^2 \)[/tex] where [tex]\( c \)[/tex] is the longest side, and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the other two sides.
First, consider each possible value for [tex]\( x \)[/tex]:
1. x = 5 inches
- [tex]\( a = 5 \)[/tex], [tex]\( b = 12 \)[/tex], [tex]\( c = 13 \)[/tex]
- Check if [tex]\( 5^2 + 12^2 < 13^2 \)[/tex]
- [tex]\( 25 + 144 < 169 \)[/tex]
- [tex]\( 169 < 169 \)[/tex] (False)
2. x = 6 inches
- [tex]\( a = 6 \)[/tex], [tex]\( b = 12 \)[/tex], [tex]\( c = 13 \)[/tex]
- Check if [tex]\( 6^2 + 12^2 < 13^2 \)[/tex]
- [tex]\( 36 + 144 < 169 \)[/tex]
- [tex]\( 180 < 169 \)[/tex] (False)
3. x = 7 inches
- [tex]\( a = 7 \)[/tex], [tex]\( b = 12 \)[/tex], [tex]\( c = 13 \)[/tex]
- Check if [tex]\( 7^2 + 12^2 < 13^2 \)[/tex]
- [tex]\( 49 + 144 < 169 \)[/tex]
- [tex]\( 193 < 169 \)[/tex] (False)
4. x = 8 inches
- [tex]\( a = 8 \)[/tex], [tex]\( b = 12 \)[/tex], [tex]\( c = 13 \)[/tex]
- Check if [tex]\( 8^2 + 12^2 < 13^2 \)[/tex]
- [tex]\( 64 + 144 < 169 \)[/tex]
- [tex]\( 208 < 169 \)[/tex] (False)
5. x = 9 inches
- [tex]\( a = 9 \)[/tex], [tex]\( b = 12 \)[/tex], [tex]\( c = 13 \)[/tex]
- Check if [tex]\( 9^2 + 12^2 < 13^2 \)[/tex]
- [tex]\( 81 + 144 < 169 \)[/tex]
- [tex]\( 225 < 169 \)[/tex] (False)
After checking all possible lengths of the shortest straw, none satisfy the condition for forming an obtuse triangle with the sides 12 inches and 13 inches.
Therefore, there are no lengths from the given options for the shortest straw that would form an obtuse triangle.
The result is:
- None of the given lengths (5 inches, 6 inches, 7 inches, 8 inches, 9 inches) are possible lengths of the shortest straw that can form an obtuse triangle with the other two sides being 12 inches and 13 inches.
Hence, the correct answer is:
[tex]\[
\boxed{}
\][/tex]
No valid lengths.
