In triangle $△abc$, $b=620$ cm, $\angle c=106°$ and $\angle a=48°$. Find the length of $a$, to the nearest centimeter.

• A. $912$ cm
• B. $453$ cm
• C. $678$ cm
• D. $1050$ cm

Answer: (A)

Verified step by step guidance

1. Identify the given elements in triangle \( \triangle abc \): side \( b = 620 \) cm, angle \( \angle c = 106^\circ \), and angle \( \angle a = 48^\circ \).
2. Calculate the remaining angle \( \angle b \) using the triangle angle sum property: \( \angle b = 180^\circ - \angle a - \angle c \).
3. Use the Law of Sines, which states that \( \frac{a}{\sin(\angle a)} = \frac{b}{\sin(\angle b)} = \frac{c}{\sin(\angle c)} \), to set up the equation to find side \( a \):
4. \[ a = b \times \frac{\sin(\angle a)}{\sin(\angle b)} \]
5. Substitute the known values of \( b \), \( \angle a \), and \( \angle b \) into the equation and solve for \( a \). Finally, round the result to the nearest centimeter.