Answer :
To determine the possible lengths of the shortest straw that form an obtuse triangle with two given straws of lengths 5 inches and 8 inches, we need to understand the properties of obtuse triangles.
An obtuse triangle has one angle greater than 90 degrees. This occurs when the square of the length of the longest side of the triangle is greater than the sum of the squares of the other two sides. Mathematically, for a triangle with side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] where [tex]\(c\)[/tex] is the longest side, this relationship is expressed as:
[tex]\[ c^2 > a^2 + b^2 \][/tex]
Let's denote the unknown length of the shortest straw as [tex]\(x\)[/tex] inches. The possible lengths provided to us are 5, 6, 7, 8, and 9 inches.
Now, let's check these values one by one:
1. When the shortest straw is 5 inches:
- Possible side lengths are 5, 5, and 8.
- Check the inequality: [tex]\(8^2 > 5^2 + 5^2\)[/tex]
- [tex]\(64 > 25 + 25\)[/tex]
- [tex]\(64 > 50\)[/tex]
Since the inequality holds, 5 inches is a possible length for the shortest straw.
2. When the shortest straw is 6 inches:
- Possible side lengths are 5, 6, and 8.
- Check the inequality: [tex]\(8^2 > 5^2 + 6^2\)[/tex]
- [tex]\(64 > 25 + 36\)[/tex]
- [tex]\(64 > 61\)[/tex]
Since the inequality holds, 6 inches is a possible length for the shortest straw.
3. When the shortest straw is 7 inches:
- Possible side lengths are 5, 7, and 8.
- Check the inequality: [tex]\(8^2 > 5^2 + 7^2\)[/tex]
- [tex]\(64 > 25 + 49\)[/tex]
- [tex]\(64 \ngtr 74\)[/tex]
Since the inequality does not hold, 7 inches is not a possible length for the shortest straw.
4. When the shortest straw is 8 inches:
- Possible side lengths are 5, 8, and 8.
- Check the inequality: [tex]\(8^2 > 5^2 + 8^2\)[/tex]
- [tex]\(64 > 25 + 64\)[/tex]
- [tex]\(64 \ngtr 89\)[/tex]
Since the inequality does not hold, 8 inches is not a possible length for the shortest straw.
5. When the shortest straw is 9 inches:
- Possible side lengths are 5, 8, and 9.
- Check the inequality: [tex]\(9^2 > 5^2 + 8^2\)[/tex]
- [tex]\(81 > 25 + 64\)[/tex]
- [tex]\(81 = 89\)[/tex]
Since the inequality does not hold, 9 inches is not a possible length for the shortest straw.
Therefore, the possible lengths for the shortest straw to form an obtuse triangle with straws of lengths 5 inches and 8 inches are:
- 5 inches
- 6 inches
An obtuse triangle has one angle greater than 90 degrees. This occurs when the square of the length of the longest side of the triangle is greater than the sum of the squares of the other two sides. Mathematically, for a triangle with side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] where [tex]\(c\)[/tex] is the longest side, this relationship is expressed as:
[tex]\[ c^2 > a^2 + b^2 \][/tex]
Let's denote the unknown length of the shortest straw as [tex]\(x\)[/tex] inches. The possible lengths provided to us are 5, 6, 7, 8, and 9 inches.
Now, let's check these values one by one:
1. When the shortest straw is 5 inches:
- Possible side lengths are 5, 5, and 8.
- Check the inequality: [tex]\(8^2 > 5^2 + 5^2\)[/tex]
- [tex]\(64 > 25 + 25\)[/tex]
- [tex]\(64 > 50\)[/tex]
Since the inequality holds, 5 inches is a possible length for the shortest straw.
2. When the shortest straw is 6 inches:
- Possible side lengths are 5, 6, and 8.
- Check the inequality: [tex]\(8^2 > 5^2 + 6^2\)[/tex]
- [tex]\(64 > 25 + 36\)[/tex]
- [tex]\(64 > 61\)[/tex]
Since the inequality holds, 6 inches is a possible length for the shortest straw.
3. When the shortest straw is 7 inches:
- Possible side lengths are 5, 7, and 8.
- Check the inequality: [tex]\(8^2 > 5^2 + 7^2\)[/tex]
- [tex]\(64 > 25 + 49\)[/tex]
- [tex]\(64 \ngtr 74\)[/tex]
Since the inequality does not hold, 7 inches is not a possible length for the shortest straw.
4. When the shortest straw is 8 inches:
- Possible side lengths are 5, 8, and 8.
- Check the inequality: [tex]\(8^2 > 5^2 + 8^2\)[/tex]
- [tex]\(64 > 25 + 64\)[/tex]
- [tex]\(64 \ngtr 89\)[/tex]
Since the inequality does not hold, 8 inches is not a possible length for the shortest straw.
5. When the shortest straw is 9 inches:
- Possible side lengths are 5, 8, and 9.
- Check the inequality: [tex]\(9^2 > 5^2 + 8^2\)[/tex]
- [tex]\(81 > 25 + 64\)[/tex]
- [tex]\(81 = 89\)[/tex]
Since the inequality does not hold, 9 inches is not a possible length for the shortest straw.
Therefore, the possible lengths for the shortest straw to form an obtuse triangle with straws of lengths 5 inches and 8 inches are:
- 5 inches
- 6 inches
