Question
Question 7
The base of an isosceles triangle is 5 inches. Which statement is true?
Multiple choice question.
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A)
The lengths of the congruent sides must be greater than 2.5 inches.
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B)
The lengths of the congruent sides must be less than 2.5 inches.
Disable choicecross out
C)
The lengths of the congruent sides must be greater than 5 inches.
Disable choicecross out
D)
The lengths of the congruent sides must be less than 5 inches.
The base of an isosceles triangle is 5 inches. Which statement is true?
Multiple choice question.
Disable choicecross out
A)
The lengths of the congruent sides must be greater than 2.5 inches.
Disable choicecross out
B)
The lengths of the congruent sides must be less than 2.5 inches.
Disable choicecross out
C)
The lengths of the congruent sides must be greater than 5 inches.
Disable choicecross out
D)
The lengths of the congruent sides must be less than 5 inches.
Answers
GPT-4o mini
To determine which statement is true for the lengths of the congruent sides in an isosceles triangle with a base of 5 inches, we can apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, consider the two congruent sides, which we'll call \( a \), and the base, which is 5 inches.
1. The triangle inequality states:
- \( a + a > 5 \) → \( 2a > 5 \) → \( a > 2.5 \)
- This means the lengths of the congruent sides must be greater than 2.5 inches.
2. We also need to consider:
- \( a + 5 > a \) (which is always true)
- \( a + 5 > a \) (which is also always true)
- However, we have another inequality derived:
- \( a + 5 > a \) (which is trivially true)
3. Lastly, it is important to note:
- \( a + a > 5 \) means \( a > 2.5 \) (already established).
- There is no restriction that states \( a \) must be less than 5 inches.
Given these considerations, let's analyze the answer choices:
A) The lengths of the congruent sides must be greater than 2.5 inches. **(True)**
B) The lengths of the congruent sides must be less than 2.5 inches. **(False)**
C) The lengths of the congruent sides must be greater than 5 inches. **(False)**
D) The lengths of the congruent sides must be less than 5 inches. **(Not necessarily true, as they can be equal to 5 inches.)**
Therefore, the correct answer is **A) The lengths of the congruent sides must be greater than 2.5 inches.**
In this case, consider the two congruent sides, which we'll call \( a \), and the base, which is 5 inches.
1. The triangle inequality states:
- \( a + a > 5 \) → \( 2a > 5 \) → \( a > 2.5 \)
- This means the lengths of the congruent sides must be greater than 2.5 inches.
2. We also need to consider:
- \( a + 5 > a \) (which is always true)
- \( a + 5 > a \) (which is also always true)
- However, we have another inequality derived:
- \( a + 5 > a \) (which is trivially true)
3. Lastly, it is important to note:
- \( a + a > 5 \) means \( a > 2.5 \) (already established).
- There is no restriction that states \( a \) must be less than 5 inches.
Given these considerations, let's analyze the answer choices:
A) The lengths of the congruent sides must be greater than 2.5 inches. **(True)**
B) The lengths of the congruent sides must be less than 2.5 inches. **(False)**
C) The lengths of the congruent sides must be greater than 5 inches. **(False)**
D) The lengths of the congruent sides must be less than 5 inches. **(Not necessarily true, as they can be equal to 5 inches.)**
Therefore, the correct answer is **A) The lengths of the congruent sides must be greater than 2.5 inches.**