To determine the value of [tex]\( a^2 \)[/tex] in an isosceles triangle with given parameters, follow these steps:
1. Identify the given values:
- Angle [tex]\( A = \frac{\pi}{6} \)[/tex]
- Sides [tex]\( b = c = 5 \)[/tex]
2. Recall the law of cosines:
The law of cosines states that for a triangle with sides [tex]\( a, b, c \)[/tex] and opposite angles [tex]\( A, B, C \)[/tex]:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
3. Substitute the given values into the law of cosines formula:
[tex]\[
a^2 = 5^2 + 5^2 - 2 \cdot 5 \cdot 5 \cdot \cos\left(\frac{\pi}{6}\right)
\][/tex]
4. Evaluate the cosine component:
[tex]\[
\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\][/tex]
5. Substitute the cosine value into the formula:
[tex]\[
a^2 = 25 + 25 - 2 \cdot 5 \cdot 5 \cdot \frac{\sqrt{3}}{2}
\][/tex]
6. Simplify the expression:
[tex]\[
a^2 = 25 + 25 - 25\sqrt{3}
\][/tex]
[tex]\[
a^2 = 50 - 25\sqrt{3}
\][/tex]
[tex]\[
a^2 = 25(2 - \sqrt{3})
\][/tex]
7. Compare this result with the provided answer choices:
- A. [tex]\( 5^2 \sqrt{3} \)[/tex]
- B. [tex]\( 5^2(\sqrt{3} - 2) \)[/tex]
- C. [tex]\( 5^2(2 - \sqrt{3}) \)[/tex]
- D. [tex]\( 5^2(2 + \sqrt{3}) \)[/tex]
8. Identify the correct answer:
The correct value matches option C:
[tex]\[
a^2 = 25(2 - \sqrt{3}) = 5^2(2 - \sqrt{3})
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{5^2(2 - \sqrt{3})} \][/tex]
