Question
Dauzat Banner
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
No Triangle
A triangle with side lengths of 5 inches, 7 inches, and 10 inches
Query 2
One and Only One Triangle
A triangle with each angle measuring 60 degrees
Query 3
One and Only One Triangle
A triangle with angle measures of
50 50 and 60
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
No Triangle
A triangle with side lengths of 5 inches, 7 inches, and 10 inches
Query 2
One and Only One Triangle
A triangle with each angle measuring 60 degrees
Query 3
One and Only One Triangle
A triangle with angle measures of
50 50 and 60
Answers
GPT-4o mini
Let's evaluate each of the triangle conditions you provided:
1. A triangle with side lengths measuring 4 inches, 5 inches, and 9 inches.
- 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. In this case, 4 + 5 is not greater than 9.
- **Query 1: No Triangle**
2. A triangle with side lengths of 5 inches, 7 inches, and 10 inches.
- This satisfies the triangle inequality theorem: 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5. Therefore, this triangle can be formed and there is only one triangle that can be constructed for these specific side lengths.
- **Query 2: One and Only One Triangle**
3. A triangle with each angle measuring 60 degrees.
- This is an equilateral triangle, and there is only one triangle that can be formed with angles measuring 60 degrees. All sides will be equal in length.
- **Query 3: One and Only One Triangle**
4. A triangle with angle measures of 50, 50, and 60.
- This is an isosceles triangle (two angles are equal), and there will be exactly one triangle with these specific angle measures.
- **Query 4: 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.
- 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. In this case, 4 + 5 is not greater than 9.
- **Query 1: No Triangle**
2. A triangle with side lengths of 5 inches, 7 inches, and 10 inches.
- This satisfies the triangle inequality theorem: 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5. Therefore, this triangle can be formed and there is only one triangle that can be constructed for these specific side lengths.
- **Query 2: One and Only One Triangle**
3. A triangle with each angle measuring 60 degrees.
- This is an equilateral triangle, and there is only one triangle that can be formed with angles measuring 60 degrees. All sides will be equal in length.
- **Query 3: One and Only One Triangle**
4. A triangle with angle measures of 50, 50, and 60.
- This is an isosceles triangle (two angles are equal), and there will be exactly one triangle with these specific angle measures.
- **Query 4: 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