1. **State the problem:** We have an isosceles triangle with base $8$ mi, height $6.7$ mi, and two equal sides $7.8$ mi each. Inside it, a circle is inscribed with radius $4.6$ mi. We want to analyze the areas involved. 2. **Formula for the area of a triangle:** $$A = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Calculate the area of the triangle:** $$A = \frac{1}{2} \times 8 \times 6.7 = 4 \times 6.7 = 26.8 \text{ mi}^2$$ 4. **Formula for the area of a circle:** $$A = \pi r^2$$ 5. **Calculate the area of the inscribed circle:** $$A = \pi \times (4.6)^2 = \pi \times 21.16 \approx 66.42 \text{ mi}^2$$ 6. **Calculate the difference between the triangle height and circle radius:** $$6.7 - 4.6 = 2.1 \text{ mi}$$ 7. **Calculate the difference between the triangle area and circle area:** $$26.8 - 66.42 = -39.62 \text{ mi}^2$$ This negative value indicates the circle's area is larger than the triangle's area, which is unusual for an inscribed circle, suggesting a possible error in the given data or interpretation. **Final answer:** - Triangle area: $26.8$ mi$^2$ - Circle area: approximately $66.42$ mi$^2$ - Height minus radius: $2.1$ mi - Area difference: $-39.62$ mi$^2$ (circle area larger than triangle area)