1. **State the problem:** Triangle DEF is formed by connecting the midpoints of the sides of triangle ABC. The sides of DEF are given as DE = 2, EF = 3, and DF = 4. We need to find the length of side BC of triangle ABC. 2. **Recall the midpoint theorem:** The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. 3. **Apply the theorem:** Since DEF is formed by midpoints, - DE is parallel to AC and DE = 1/2 AC - EF is parallel to AB and EF = 1/2 AB - DF is parallel to BC and DF = 1/2 BC 4. **Find BC:** Given DF = 4, and DF = 1/2 BC, $$4 = \frac{1}{2} BC$$ Multiply both sides by 2: $$2 \times 4 = \cancel{2} \times \frac{1}{\cancel{2}} BC \Rightarrow 8 = BC$$ 5. **Answer:** The length of BC is 8 units. --- Since the user asked two questions, the second is: 1. **State the problem:** If AB = 12, find the length of A'B'. 2. **Interpretation:** A' and B' are points dividing sides AB and BC respectively, but from the description, it seems A' and B' are points on AB and BC such that segments are divided as A to A' = 2, A' to C = 4, B to B' = 3, B' to C = 6. However, since the question is about A'B', and AB = 12, we consider the scale. 3. **Since A' and B' are midpoints or points dividing sides proportionally, and AB = 12, the length A'B' corresponds to half or a fraction of AB depending on the figure. Without additional info, assuming A'B' is half of AB (midpoint segment), then: $$A'B' = \frac{1}{2} AB = \frac{1}{2} \times 12 = 6$$ 4. **Answer:** The length of A'B' is 6 units.