Question
which side of measures will create many triangles
15 m, 13 m, 18 m
10 m, 10 m, 10 m
65°, 5°, 110°,
91°, 54°, 34°
15 m, 13 m, 18 m
10 m, 10 m, 10 m
65°, 5°, 110°,
91°, 54°, 34°
Answers
GPT-4o mini
To determine which set of measures will create many triangles, we can use the properties of triangles, specifically the triangle inequality for lengths and the triangle angle sum property for angles.
1. **For the lengths:**
- The triangle inequality theorem states that for any triangle with sides \( a \), \( b \), and \( c \), the following inequalities must be true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's check the lengths given:
- **15 m, 13 m, 18 m:**
- \( 15 + 13 = 28 > 18 \) (true)
- \( 15 + 18 = 33 > 13 \) (true)
- \( 13 + 18 = 31 > 15 \) (true)
- This set can form a triangle.
- **10 m, 10 m, 10 m:**
- \( 10 + 10 = 20 > 10 \) (true)
- This set can form a triangle as well.
2. **For the angles:**
- The triangle sum theorem states that the sum of the angles in a triangle must be 180°.
Let's check the angles given:
- **65°, 5°, 110°:**
- \( 65 + 5 + 110 = 180° \) (true)
- This set can form a triangle.
- **91°, 54°, 34°:**
- \( 91 + 54 + 34 = 179° \) (not equal to 180°)
- This set cannot form a triangle.
Now, let's summarize:
- The lengths \( 15 m, 13 m, 18 m \) can form a triangle.
- The lengths \( 10 m, 10 m, 10 m \) can also form an equilateral triangle.
- The angles \( 65°, 5°, 110° \) can form a triangle.
- The angles \( 91°, 54°, 34° \) cannot form a triangle.
To create many triangles, we would need to consider sets that can form multiple combinations or subsets. The sets of lengths \( (15 m, 13 m, 18 m) \) and \( (10 m, 10 m, 10 m) \) are valid. However, based on the character of the measures provided, the lengths \( 10 m, 10 m, 10 m \) stand out because they create an equilateral triangle which has many congruent triangles formed by dividing it into smaller triangles.
Conclusively:
- **(10 m, 10 m, 10 m)** will create many congruent triangles (as it is equilateral, and all smaller triangles would be congruent).
- There are fewer configurations for the other measures, and one set of angles cannot create a triangle at all.
Thus, if we are looking for variety and the potential for many triangles, **the lengths 10 m, 10 m, 10 m** is the most interesting choice.
1. **For the lengths:**
- The triangle inequality theorem states that for any triangle with sides \( a \), \( b \), and \( c \), the following inequalities must be true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's check the lengths given:
- **15 m, 13 m, 18 m:**
- \( 15 + 13 = 28 > 18 \) (true)
- \( 15 + 18 = 33 > 13 \) (true)
- \( 13 + 18 = 31 > 15 \) (true)
- This set can form a triangle.
- **10 m, 10 m, 10 m:**
- \( 10 + 10 = 20 > 10 \) (true)
- This set can form a triangle as well.
2. **For the angles:**
- The triangle sum theorem states that the sum of the angles in a triangle must be 180°.
Let's check the angles given:
- **65°, 5°, 110°:**
- \( 65 + 5 + 110 = 180° \) (true)
- This set can form a triangle.
- **91°, 54°, 34°:**
- \( 91 + 54 + 34 = 179° \) (not equal to 180°)
- This set cannot form a triangle.
Now, let's summarize:
- The lengths \( 15 m, 13 m, 18 m \) can form a triangle.
- The lengths \( 10 m, 10 m, 10 m \) can also form an equilateral triangle.
- The angles \( 65°, 5°, 110° \) can form a triangle.
- The angles \( 91°, 54°, 34° \) cannot form a triangle.
To create many triangles, we would need to consider sets that can form multiple combinations or subsets. The sets of lengths \( (15 m, 13 m, 18 m) \) and \( (10 m, 10 m, 10 m) \) are valid. However, based on the character of the measures provided, the lengths \( 10 m, 10 m, 10 m \) stand out because they create an equilateral triangle which has many congruent triangles formed by dividing it into smaller triangles.
Conclusively:
- **(10 m, 10 m, 10 m)** will create many congruent triangles (as it is equilateral, and all smaller triangles would be congruent).
- There are fewer configurations for the other measures, and one set of angles cannot create a triangle at all.
Thus, if we are looking for variety and the potential for many triangles, **the lengths 10 m, 10 m, 10 m** is the most interesting choice.