Question
A triangle has two side lengths of 10 and 16 units. The third side is half the length of one of the sides. How many possible triangles can be formed given these conditions? Explain.
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A.
No triangles can be formed. The sum of the lengths of the two shorter sides are both less than the length of the longest side.
B.
Two triangles can be formed. The possible lengths of the third side are
enter your response here and
enter your response here units. The sum of the lengths of the two shorter sides are both greater than the length of the longest side.
(Type integers or decimals. Use ascending order.)
C.
One triangle can be formed. The length of the third side is
enter your response here units. The sum of the lengths of the two shorter sides is greater than the length of the longest side.
(Type an integer or a decimal.)
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A.
No triangles can be formed. The sum of the lengths of the two shorter sides are both less than the length of the longest side.
B.
Two triangles can be formed. The possible lengths of the third side are
enter your response here and
enter your response here units. The sum of the lengths of the two shorter sides are both greater than the length of the longest side.
(Type integers or decimals. Use ascending order.)
C.
One triangle can be formed. The length of the third side is
enter your response here units. The sum of the lengths of the two shorter sides is greater than the length of the longest side.
(Type an integer or a decimal.)
Answers
Answered by
GPT-4o mini
To solve the problem, we need to determine the possible lengths for the third side of the triangle, which is stated to be half the length of one of the other sides.
Given the lengths of the two sides are 10 and 16, we have the following possible lengths for the third side:
1. **Half the length of the first side (10 units):**
\[
\text{Third side} = \frac{10}{2} = 5 \text{ units}
\]
2. **Half the length of the second side (16 units):**
\[
\text{Third side} = \frac{16}{2} = 8 \text{ units}
\]
Now, we need to determine if these lengths (5 and 8 units) can form valid triangles with the other two sides (10 and 16 units). For a set of three side lengths to form a triangle, they must satisfy the triangle inequality theorem which states that the sum of the lengths of any two sides must be greater than the length of the third side.
Let's check each case:
### Case 1: Third side = 5 units
- **Sides:** 5, 10, 16
- \(5 + 10 = 15 > 16\) (False)
- \(5 + 16 = 21 > 10\) (True)
- \(10 + 16 = 26 > 5\) (True)
Since one of the inequalities fails, a triangle cannot be formed with side lengths 5, 10, and 16.
### Case 2: Third side = 8 units
- **Sides:** 8, 10, 16
- \(8 + 10 = 18 > 16\) (True)
- \(8 + 16 = 24 > 10\) (True)
- \(10 + 16 = 26 > 8\) (True)
All inequalities are satisfied, so a triangle can indeed be formed with side lengths 8, 10, and 16.
### Conclusion
Based on the evaluations, we find:
- No triangle can be formed with a third side of 5 units.
- A triangle can be formed with a third side of 8 units.
Thus, it is confirmed that **only one triangle can be formed with the side length of 8 units**. The sum of the lengths of the two shorter sides (8 and 10, for example) is greater than the length of the longest side (16).
Therefore, the correct choice is:
C. One triangle can be formed. The length of the third side is **8 units**. The sum of the lengths of the two shorter sides is greater than the length of the longest side.
Given the lengths of the two sides are 10 and 16, we have the following possible lengths for the third side:
1. **Half the length of the first side (10 units):**
\[
\text{Third side} = \frac{10}{2} = 5 \text{ units}
\]
2. **Half the length of the second side (16 units):**
\[
\text{Third side} = \frac{16}{2} = 8 \text{ units}
\]
Now, we need to determine if these lengths (5 and 8 units) can form valid triangles with the other two sides (10 and 16 units). For a set of three side lengths to form a triangle, they must satisfy the triangle inequality theorem which states that the sum of the lengths of any two sides must be greater than the length of the third side.
Let's check each case:
### Case 1: Third side = 5 units
- **Sides:** 5, 10, 16
- \(5 + 10 = 15 > 16\) (False)
- \(5 + 16 = 21 > 10\) (True)
- \(10 + 16 = 26 > 5\) (True)
Since one of the inequalities fails, a triangle cannot be formed with side lengths 5, 10, and 16.
### Case 2: Third side = 8 units
- **Sides:** 8, 10, 16
- \(8 + 10 = 18 > 16\) (True)
- \(8 + 16 = 24 > 10\) (True)
- \(10 + 16 = 26 > 8\) (True)
All inequalities are satisfied, so a triangle can indeed be formed with side lengths 8, 10, and 16.
### Conclusion
Based on the evaluations, we find:
- No triangle can be formed with a third side of 5 units.
- A triangle can be formed with a third side of 8 units.
Thus, it is confirmed that **only one triangle can be formed with the side length of 8 units**. The sum of the lengths of the two shorter sides (8 and 10, for example) is greater than the length of the longest side (16).
Therefore, the correct choice is:
C. One triangle can be formed. The length of the third side is **8 units**. The sum of the lengths of the two shorter sides is greater than the length of the longest side.