To determine what must be true for the angle opposite the side of length [tex]\(a\)[/tex] to be acute in a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we need to utilize properties of triangles and their angles.
When an angle in a triangle is acute, the following relationship between the sides must hold:
[tex]\[ b^2 + c^2 > a^2 \][/tex]
Here's the step-by-step solution:
1. Understand the nature of the question: We need to determine which side inequality ensures that the angle opposite [tex]\(a\)[/tex] is acute. In this context, "acute" means less than 90 degrees.
2. Properties of a right triangle:
- For a right triangle, the Pythagorean theorem states [tex]\( a^2 + b^2 = c^2 \)[/tex] where [tex]\(c\)[/tex] is the hypotenuse (longest side).
- If [tex]\( a^2 + b^2 < c^2 \)[/tex], then the triangle has an obtuse angle.
3. Property of acute angles in triangles:
- For any triangle with an acute angle, the square of one side must be less than the sum of the squares of the other two sides.
4. Application to given sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- To ensure the angle opposite [tex]\(a\)[/tex] is acute, [tex]\( a \neq \)[/tex] hypotenuse and the inequality [tex]\( b^2 + c^2 > a^2 \)[/tex] must hold true.
So, the correct condition is:
[tex]\[ \boxed{b^2 + c^2 > a^2} \][/tex]
Thus, the answer is Option C: [tex]\( b^2 + c^2 > a^2 \)[/tex].
