Given triangle $△\mathrm{qrs}$ is similar to triangle $△\mathrm{xyz}$, which of the following ratios is equal according to the Law of Sines?

• A. $\frac{s}{sin\left(Y\right)}=\frac{z}{sin\left(R\right)}$
• B. $\frac{r}{sin\left(Z\right)}=\frac{y}{sin\left(Q\right)}$
• C. $\frac{q}{sin\left(Q\right)}=\frac{x}{sin\left(X\right)}$
• D. $\frac{q}{sin\left(R\right)}=\frac{x}{sin\left(Y\right)}$

Answer: (C)

Verified step by step guidance

1. Recall that the Law of Sines states for any triangle ABC: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\), where \(a\), \(b\), and \(c\) are the sides opposite angles \(A\), \(B\), and \(C\) respectively.
2. Since triangles \(\triangle qrs\) and \(\triangle xyz\) are similar, their corresponding angles are equal: \(Q\) corresponds to \(X\), \(R\) corresponds to \(Y\), and \(S\) corresponds to \(Z\).
3. Identify the sides opposite each angle in both triangles: side \(q\) is opposite angle \(Q\), side \(r\) opposite \(R\), side \(s\) opposite \(S\); similarly, side \(x\) opposite \(X\), \(y\) opposite \(Y\), and \(z\) opposite \(Z\).
4. Apply the Law of Sines to each triangle separately, then set the ratios of corresponding sides and sines of their opposite angles equal due to similarity: \(\frac{q}{\sin(Q)} = \frac{x}{\sin(X)}\), \(\frac{r}{\sin(R)} = \frac{y}{\sin(Y)}\), and \(\frac{s}{\sin(S)} = \frac{z}{\sin(Z)}\).
5. From these equalities, select the ratio that matches the given correct answer: \(\frac{q}{\sin(Q)} = \frac{x}{\sin(X)}\).