Question
Sam is drawing triangles on banners using a ruler and protractor.
Inline Dropdown Question
On the second banner Sam wants to create a variety of different sizes of triangles.
For each triangle condition listed, determine whether the condition will produce one and only one triangle, more than one triangle, or no triangle.
Use the drop-down menu to select the appropriate answer.
A triangle with side lengths measuring 4 inches, 5 inches, and 9 inches
Query 1
A triangle with side lengths of 5 inches, 7 inches, and 10 inches
Query 2
A triangle with each angle measuring 60 degrees
Query 3
A triangle with angle measures of 50, 50, and
60
Query 4
Answers
GPT-4o mini
Let's analyze each triangle condition:
1. **A triangle with side lengths measuring 4 inches, 5 inches, and 9 inches**
- The sum of the lengths of the two shorter sides (4 + 5 = 9) equals the length of the longest side (9). This does not 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.
- **Answer**: No triangle
2. **A triangle with side lengths of 5 inches, 7 inches, and 10 inches**
- The sum of the lengths of the two shorter sides (5 + 7 = 12) is greater than the length of the longest side (10). It satisfies the triangle inequality theorem.
- **Answer**: One and only one triangle
3. **A triangle with each angle measuring 60 degrees**
- A triangle with all angles measuring 60 degrees is an equilateral triangle. There is only one unique shape that satisfies this condition as all sides will be equal.
- **Answer**: One and only one triangle
4. **A triangle with angle measures of 50, 50, and 60**
- The angle measures add up to 160 degrees (50 + 50 + 60), which is valid for a triangle, and since two angles are the same, it will be an isosceles triangle.
- **Answer**: One and only one triangle
To summarize:
- Query 1: No triangle
- Query 2: One and only one triangle
- Query 3: One and only one triangle
- Query 4: One and only one triangle
1. **A triangle with side lengths measuring 4 inches, 5 inches, and 9 inches**
- The sum of the lengths of the two shorter sides (4 + 5 = 9) equals the length of the longest side (9). This does not 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.
- **Answer**: No triangle
2. **A triangle with side lengths of 5 inches, 7 inches, and 10 inches**
- The sum of the lengths of the two shorter sides (5 + 7 = 12) is greater than the length of the longest side (10). It satisfies the triangle inequality theorem.
- **Answer**: One and only one triangle
3. **A triangle with each angle measuring 60 degrees**
- A triangle with all angles measuring 60 degrees is an equilateral triangle. There is only one unique shape that satisfies this condition as all sides will be equal.
- **Answer**: One and only one triangle
4. **A triangle with angle measures of 50, 50, and 60**
- The angle measures add up to 160 degrees (50 + 50 + 60), which is valid for a triangle, and since two angles are the same, it will be an isosceles triangle.
- **Answer**: One and only one triangle
To summarize:
- Query 1: No triangle
- Query 2: One and only one triangle
- Query 3: One and only one triangle
- Query 4: One and only one triangle