1. **Problem Statement:** A farmer wants to divide a triangular field into four equal smaller triangles using fences. We need to use the midline theorem to determine where to place the fences and calculate their total length, given only the perimeter of the original triangle. 2. **Understanding the Midline Theorem:** The midline theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. 3. **Step-by-step solution:** - Let the original triangle be $\triangle ABC$ with sides $a$, $b$, and $c$. - The perimeter $P = a + b + c$ is given. - Find the midpoints of each side: $M$ on $AB$, $N$ on $BC$, and $O$ on $CA$. - Connect these midpoints to form the medial triangle $\triangle MNO$. - By the midline theorem, each side of $\triangle MNO$ is half the length of the corresponding side of $\triangle ABC$. - The medial triangle divides the original triangle into four smaller triangles of equal area. 4. **Fence placement:** - The fences should be placed along the segments $MN$, $NO$, and $OM$ connecting the midpoints. 5. **Calculating total fence length:** - Each fence segment is half the length of the original side it parallels. - Total fence length $= \frac{a}{2} + \frac{b}{2} + \frac{c}{2} = \frac{a + b + c}{2} = \frac{P}{2}$. **Final answer:** The fences should be placed along the three mid-segments connecting the midpoints of the sides, and the total length of the fences required is exactly half the perimeter of the original triangular field, i.e., $$\text{Total fence length} = \frac{P}{2}.$$