Suppose an isosceles triangle [tex]\(ABC\)[/tex] has [tex]\(\angle A = \frac{\pi}{6}\)[/tex] and [tex]\(b = c = 5\)[/tex]. What is the length of [tex]\(a^2\)[/tex]?

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]



Answer :

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]