To determine which condition must be true for a triangle where the angle opposite the side of length [tex]\( a \)[/tex] is acute, we need to understand the relationships between the sides of the triangle when the angle is acute.
If the angle opposite side [tex]\( a \)[/tex] is acute, the triangle must follow certain geometric properties. Specifically, for an angle to be acute, the square of the length of the side opposite the angle (in this case, [tex]\(a\)[/tex]) must be less than the sum of the squares of the other two sides.
This can be confirmed using the Pythagorean theorem and its extensions for acute, right, and obtuse triangles:
1. For a right triangle, with the angle being [tex]\( 90^\circ \)[/tex]:
[tex]\[
a^2 = b^2 + c^2
\][/tex]
2. For an obtuse triangle, where the angle is greater than [tex]\( 90^\circ \)[/tex]:
[tex]\[
a^2 > b^2 + c^2
\][/tex]
3. For an acute triangle, where all angles are less than [tex]\( 90^\circ \)[/tex]:
[tex]\[
a^2 < b^2 + c^2
\][/tex]
Given this understanding, for a triangle with an acute angle opposite the side [tex]\( a \)[/tex], the correct condition would be:
### [tex]\( b^2 + c^2 > a^2 \)[/tex]
Thus, the correct choice is:
A. [tex]\( b^2 + c^2 > a^2 \)[/tex]
