Question

In a $\triangle ABC$, $a = 1$, $b = \sqrt{3}$ and $\angle C = \dfrac{\pi}{6}$. Then the measure of the third side $c =$}

• A. 4
• B. 3
• C. 1
• D. 2

Hint:

When two sides and the included angle are known (SAS), the Cosine Rule is the direct and only formula needed to find the third side.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves solving a triangle when two sides and the included angle (SAS property) are known.
The Law of Cosines is the standard tool to find the third side in such scenarios.
Key Formula or Approach:
1. Law of Cosines: \( c^2 = a^2 + b^2 - 2ab \cos C \).
2. Standard Value: \( \cos(\pi/6) = \cos 30^\circ = \frac{\sqrt{3}}{2} \).
Step 2: Detailed Explanation:
Given values are \( a = 1 \), \( b = \sqrt{3} \), and \( C = \pi/6 \).
Plug these values into the formula:
\[ c^2 = (1)^2 + (\sqrt{3})^2 - 2(1)(\sqrt{3}) \cos(30^\circ) \] \[ c^2 = 1 + 3 - 2\sqrt{3} \left( \frac{\sqrt{3}}{2} \right) \] The factor 2 in the numerator and denominator cancels out:
\[ c^2 = 4 - (\sqrt{3} \cdot \sqrt{3}) \] \[ c^2 = 4 - 3 = 1 \] Taking the square root (since length must be positive):
\[ c = \sqrt{1} = 1 \] Step 3: Final Answer:
The third side \( c \) has a measure of 1.
This matches Option (C).