Given that triangle $ABC$ is similar to triangle $DEF$, which of the following correctly expresses the Law of Sines for these triangles?

• A. $\frac{\sin A}{\sin B}=\frac{a}{b}$
• B. $\frac{a}{b}=\frac{\sin A}{\sin B}$
• C. $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
• D. $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Answer: (C)

Verified step by step guidance

1. Recall the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, this is expressed as \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where \(a\), \(b\), and \(c\) are the sides opposite angles \(A\), \(B\), and \(C\) respectively.
2. Since triangles \(ABC\) and \(DEF\) are similar, their corresponding angles are equal, and their corresponding sides are proportional. This means \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\), and the ratios of corresponding sides are equal.
3. Apply the Law of Sines to triangle \(ABC\) to write \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), and similarly for triangle \(DEF\) as \(\frac{d}{\sin D} = \frac{e}{\sin E} = \frac{f}{\sin F}\).
4. Because the triangles are similar, the ratios of corresponding sides are equal, so \(\frac{a}{d} = \frac{b}{e} = \frac{c}{f}\). Combining this with the Law of Sines, the equality \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\) holds true for triangle \(ABC\).
5. Therefore, the correct expression of the Law of Sines for these triangles is \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\), which shows the ratio of the sine of an angle to the length of its opposite side is constant.