1. The problem states that we have two similar triangles with areas 140.8 cm² and 178.2 cm², and we need to find the ratio of their corresponding medians. 2. For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding medians (or any corresponding linear measures). If we let the ratio of the medians be $\frac{m_1}{m_2}$, then: $$\frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{m_1}{m_2}\right)^2$$ 3. Substitute the given areas: $$\frac{140.8}{178.2} = \left(\frac{m_1}{m_2}\right)^2$$ 4. Calculate the ratio of the areas: $$\frac{140.8}{178.2} \approx 0.7901$$ 5. Take the square root of both sides to find the ratio of the medians: $$\frac{m_1}{m_2} = \sqrt{0.7901} \approx 0.889$$ 6. Therefore, the ratio of the corresponding medians is approximately $0.889:1$, or equivalently: $$\boxed{0.889:1}$$