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Suppose a triangle has sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], and that [tex]\(a^2 + b^2 \ \textless \ c^2\)[/tex]. Let [tex]\(\theta\)[/tex] be the measure of the angle opposite the side of length [tex]\(c\)[/tex]. Which of the following must be true? Check all that apply.

A. [tex]\(a^2 + b^2 - c^2 = 2ab \cos \theta\)[/tex]
B. [tex]\(\cos \theta \ \textgreater \ 0\)[/tex]
C. The triangle is not a right triangle.
D. [tex]\(\cos \theta \ \textless \ 0\)[/tex]

Sagot :

GinnyAnswer
Given that [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 [tex]\(c\)[/tex], we need to determine which of the following statements are true:

A. [tex]\(a^2 + b^2 - c^2 = 2 a b \cos \theta\)[/tex]

B. [tex]\(\cos \theta > 0\)[/tex]

C. The triangle is not a right triangle.

D. [tex]\(\cos \theta < 0\)[/tex]

Step-by-Step Analysis:

### 1. Analyzing Statement A
The equation [tex]\(a^2 + b^2 - c^2 = 2 a b \cos \theta\)[/tex] is derived from the Law of Cosines, which states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos \theta \][/tex]
Rewriting the equation, we have:
[tex]\[ a^2 + b^2 - c^2 = 2ab \cos \theta \][/tex]
This statement holds true regardless of the relationship among [tex]\(a^2\)[/tex], [tex]\(b^2\)[/tex], and [tex]\(c^2\)[/tex]. Therefore, statement A is generally true, but without specific context, it doesn't necessarily help us conclude anything about the specific nature of the triangle in this context.

### 2. Analyzing Statement B
The cosine of an angle in a triangle depends on the relationship between the side lengths. Given [tex]\(a^2 + b^2 < c^2\)[/tex], the triangle must be obtuse, as in an obtuse triangle, the square of one side is greater than the sum of the squares of the other two sides. In an obtuse triangle, the cosine of the angle opposite the longest side is negative:
[tex]\[ \cos \theta < 0 \][/tex]
Given that [tex]\(\cos \theta\)[/tex] must be negative in this situation, statement B is false.

### 3. Analyzing Statement C
In a right triangle, the Pythagorean theorem holds, so [tex]\(a^2 + b^2 = c^2\)[/tex]. The given condition [tex]\(a^2 + b^2 < c^2\)[/tex] violates this theorem, so the triangle cannot be a right triangle. Therefore, statement C is true.

### 4. Analyzing Statement D
As analyzed in Statement B, for an obtuse triangle where [tex]\(a^2 + b^2 < c^2\)[/tex], the cosine of the angle opposite the longest side is negative, thus:
[tex]\[ \cos \theta < 0 \][/tex]
Therefore, statement D is true.

### Conclusion
Based on the analysis:
- Statement C is true.
- Statement D is true.

Thus, the correct options are [3, 4].
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