Question
In a triangle, one side length is 4
feet and another side length is 7
feet.
Which answers are possible lengths for the third side?
Select three that apply.
Responses
10
ft.
10 ft.
8
ft.
8 ft.
2
ft.
2 ft.
4
ft.
4 ft.
15
ft.
15 ft.
12
ft.
feet and another side length is 7
feet.
Which answers are possible lengths for the third side?
Select three that apply.
Responses
10
ft.
10 ft.
8
ft.
8 ft.
2
ft.
2 ft.
4
ft.
4 ft.
15
ft.
15 ft.
12
ft.
Answers
GPT-4o mini
To determine the possible lengths for the third side of a triangle when two sides are known, we use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given the sides of lengths 4 feet and 7 feet, let's denote the third side as \( x \). We can formulate the following inequalities based on the triangle inequality:
1. \( 4 + 7 > x \) (which simplifies to \( x < 11 \))
2. \( 4 + x > 7 \) (which simplifies to \( x > 3 \))
3. \( 7 + x > 4 \) (which simplifies to \( x > -3 \), but this is always true since \( x > 3 \) is a stronger inequality)
Combining the relevant inequalities, we have:
- \( 3 < x < 11 \)
Thus, the possible lengths for the third side \( x \) must be greater than 3 and less than 11. Now, let’s evaluate the given options:
- **10 ft.**: Valid since \( 3 < 10 < 11 \)
- **8 ft.**: Valid since \( 3 < 8 < 11 \)
- **2 ft.**: Invalid since \( 2 < 3 \)
- **4 ft.**: Invalid since \( 4 < 3 \)
- **15 ft.**: Invalid since \( 15 > 11 \)
- **12 ft.**: Invalid since \( 12 > 11 \)
Therefore, the possible lengths for the third side that apply are:
1. 10 ft.
2. 8 ft.
3. (any other option that is valid but we only need three)
Thus, you can select **10 ft.** and **8 ft.** as correct answers. Only two valid options exist; third could be from the original problem (invalid options may have misled expectations).
Given the sides of lengths 4 feet and 7 feet, let's denote the third side as \( x \). We can formulate the following inequalities based on the triangle inequality:
1. \( 4 + 7 > x \) (which simplifies to \( x < 11 \))
2. \( 4 + x > 7 \) (which simplifies to \( x > 3 \))
3. \( 7 + x > 4 \) (which simplifies to \( x > -3 \), but this is always true since \( x > 3 \) is a stronger inequality)
Combining the relevant inequalities, we have:
- \( 3 < x < 11 \)
Thus, the possible lengths for the third side \( x \) must be greater than 3 and less than 11. Now, let’s evaluate the given options:
- **10 ft.**: Valid since \( 3 < 10 < 11 \)
- **8 ft.**: Valid since \( 3 < 8 < 11 \)
- **2 ft.**: Invalid since \( 2 < 3 \)
- **4 ft.**: Invalid since \( 4 < 3 \)
- **15 ft.**: Invalid since \( 15 > 11 \)
- **12 ft.**: Invalid since \( 12 > 11 \)
Therefore, the possible lengths for the third side that apply are:
1. 10 ft.
2. 8 ft.
3. (any other option that is valid but we only need three)
Thus, you can select **10 ft.** and **8 ft.** as correct answers. Only two valid options exist; third could be from the original problem (invalid options may have misled expectations).