1. **Problem statement:** We have a triangle ABC with a point D on segment AB such that $|AD|=1$ and $|BD|=2$. The segment CD is perpendicular to AB, dividing the triangle into 7 pieces with areas labeled 4, 5, 5, $x$, 3, 1, and 8. We need to find the area $x$. 2. **Key idea:** Since CD is perpendicular to AB, the areas of the smaller triangles formed with base segments on AB are proportional to the lengths of those base segments. 3. **Step 1: Understand the division of AB.** - $|AD|=1$ - $|DB|=2$ - So, $|AB|=3$ 4. **Step 2: Use the areas given to find the height from C to AB.** - The total area of triangle ABC is the sum of all 7 pieces: $$4 + 5 + 5 + x + 3 + 1 + 8 = 26 + x$$ 5. **Step 3: Group areas by their base segments.** - The pieces on segment AD (length 1) have areas: 4, 5, 5 - The pieces on segment DB (length 2) have areas: $x$, 3, 1, 8 6. **Step 4: Calculate total area on each segment.** - Sum on AD side: $4 + 5 + 5 = 14$ - Sum on DB side: $x + 3 + 1 + 8 = x + 12$ 7. **Step 5: Use proportionality of areas to base lengths.** - Since height is the same for both parts, area is proportional to base length: $$\frac{\text{Area on AD}}{\text{Area on DB}} = \frac{|AD|}{|DB|} = \frac{1}{2}$$ 8. **Step 6: Set up the equation and solve for $x$.** $$\frac{14}{x + 12} = \frac{1}{2}$$ Multiply both sides by $2(x + 12)$: $$2 \times 14 = x + 12$$ $$28 = x + 12$$ $$x = 28 - 12 = 16$$ 9. **Final answer:** $$\boxed{16}$$