1. The problem involves understanding the ratio of three fractions: $\frac{5}{7}$, $\frac{5\sqrt{3}}{7}$, and $\frac{10}{7}$, which are related to angles $45^\circ$, $-45^\circ$, $-90^\circ$, $30^\circ$, $-60^\circ$, and $-90^\circ$. 2. These fractions likely represent side lengths or ratios in a trigonometric context, possibly related to special triangles or trigonometric function values. 3. To analyze the ratio, we can express all fractions with a common denominator and compare their values: $$\frac{5}{7}, \quad \frac{5\sqrt{3}}{7}, \quad \frac{10}{7}$$ 4. Since the denominators are the same, compare the numerators: - $5$ - $5\sqrt{3} \approx 5 \times 1.732 = 8.66$ - $10$ 5. The ratio of the numerators is approximately $5 : 8.66 : 10$. 6. Simplify this ratio by dividing all terms by 5: $$1 : \frac{8.66}{5} : 2 = 1 : 1.732 : 2$$ 7. This ratio $1 : \sqrt{3} : 2$ corresponds to the side lengths of a $30^\circ$-$60^\circ$-$90^\circ$ triangle. 8. Therefore, the fractions represent the side ratios of this special right triangle, confirming the connection to the given angles. Final answer: The fractions $\frac{5}{7} : \frac{5\sqrt{3}}{7} : \frac{10}{7}$ simplify to the ratio $1 : \sqrt{3} : 2$, which are the side lengths of a $30^\circ$-$60^\circ$-$90^\circ$ triangle.