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Given the condition [tex]\(a^2 + b^2 < c^2\)[/tex] for a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], and [tex]\(\theta\)[/tex] being the angle opposite the side with length [tex]\(c\)[/tex], we will analyze which of the statements must be true.
### Statement A: The triangle is a right triangle.
For a triangle to be a right triangle, the Pythagorean theorem must hold, which states that [tex]\(a^2 + b^2 = c^2\)[/tex] where [tex]\(c\)[/tex] is the hypotenuse. Since we are given [tex]\(a^2 + b^2 < c^2\)[/tex], the condition for a right triangle is not satisfied. Therefore, the triangle cannot be a right triangle.
Conclusion: This statement is false.
### Statement B: [tex]\(\theta\)[/tex] is an obtuse angle.
In any triangle, the relationship between the sides and the angles is governed by the Law of Cosines, which states:
[tex]\[ c^2 = a^2 + b^2 - 2ab\cos\theta \][/tex]
Given [tex]\(a^2 + b^2 < c^2\)[/tex], we can rewrite this as:
[tex]\[ c^2 > a^2 + b^2 \][/tex]
From the Law of Cosines:
[tex]\[ c^2 = a^2 + b^2 - 2ab\cos\theta \][/tex]
[tex]\[ a^2 + b^2 - 2ab\cos\theta = c^2 \][/tex]
Given [tex]\(a^2 + b^2 < c^2\)[/tex], it implies:
[tex]\[ a^2 + b^2 - 2ab\cos\theta < a^2 + b^2 \][/tex]
[tex]\[ - 2ab\cos\theta < 0 \][/tex]
[tex]\[ \cos\theta < 0 \][/tex]
The cosine of an angle is negative when the angle is obtuse (i.e., between 90° and 180°).
Conclusion: This statement is true.
### Statement C: [tex]\(\cos \theta < 0\)[/tex]
As derived above, [tex]\( \cos\theta < 0 \)[/tex] is true if [tex]\( a^2 + b^2 < c^2 \)[/tex]. Therefore, [tex]\(\theta\)[/tex] must be an obtuse angle since [tex]\(\cos\theta\)[/tex] is negative.
Conclusion: This statement is true.
### Statement D: The triangle is not a right triangle.
From Statement A, we concluded that the triangle cannot be a right triangle because the condition [tex]\(a^2 + b^2 = c^2\)[/tex] is not satisfied. Therefore, it is clear that the triangle is not a right triangle.
Conclusion: This statement is true.
### Summary
Given the condition [tex]\(a^2 + b^2 < c^2\)[/tex], the statements that must be true are:
- B. [tex]\(\theta\)[/tex] is an obtuse angle.
- C. [tex]\(\cos \theta < 0\)[/tex].
- D. The triangle is not a right triangle.
Thus, the correct answers are:
[tex]\[ \text{False, True, True, True} \][/tex]
### Statement A: The triangle is a right triangle.
For a triangle to be a right triangle, the Pythagorean theorem must hold, which states that [tex]\(a^2 + b^2 = c^2\)[/tex] where [tex]\(c\)[/tex] is the hypotenuse. Since we are given [tex]\(a^2 + b^2 < c^2\)[/tex], the condition for a right triangle is not satisfied. Therefore, the triangle cannot be a right triangle.
Conclusion: This statement is false.
### Statement B: [tex]\(\theta\)[/tex] is an obtuse angle.
In any triangle, the relationship between the sides and the angles is governed by the Law of Cosines, which states:
[tex]\[ c^2 = a^2 + b^2 - 2ab\cos\theta \][/tex]
Given [tex]\(a^2 + b^2 < c^2\)[/tex], we can rewrite this as:
[tex]\[ c^2 > a^2 + b^2 \][/tex]
From the Law of Cosines:
[tex]\[ c^2 = a^2 + b^2 - 2ab\cos\theta \][/tex]
[tex]\[ a^2 + b^2 - 2ab\cos\theta = c^2 \][/tex]
Given [tex]\(a^2 + b^2 < c^2\)[/tex], it implies:
[tex]\[ a^2 + b^2 - 2ab\cos\theta < a^2 + b^2 \][/tex]
[tex]\[ - 2ab\cos\theta < 0 \][/tex]
[tex]\[ \cos\theta < 0 \][/tex]
The cosine of an angle is negative when the angle is obtuse (i.e., between 90° and 180°).
Conclusion: This statement is true.
### Statement C: [tex]\(\cos \theta < 0\)[/tex]
As derived above, [tex]\( \cos\theta < 0 \)[/tex] is true if [tex]\( a^2 + b^2 < c^2 \)[/tex]. Therefore, [tex]\(\theta\)[/tex] must be an obtuse angle since [tex]\(\cos\theta\)[/tex] is negative.
Conclusion: This statement is true.
### Statement D: The triangle is not a right triangle.
From Statement A, we concluded that the triangle cannot be a right triangle because the condition [tex]\(a^2 + b^2 = c^2\)[/tex] is not satisfied. Therefore, it is clear that the triangle is not a right triangle.
Conclusion: This statement is true.
### Summary
Given the condition [tex]\(a^2 + b^2 < c^2\)[/tex], the statements that must be true are:
- B. [tex]\(\theta\)[/tex] is an obtuse angle.
- C. [tex]\(\cos \theta < 0\)[/tex].
- D. The triangle is not a right triangle.
Thus, the correct answers are:
[tex]\[ \text{False, True, True, True} \][/tex]
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