\( \leftarrow \begin{array}{l}\text { A vacant city lot is being turned into a neighborhood garden. The neighbors want to fence in a triangular section of the lot and plant flowers there. The longest side of the triangular section is } 7 \text { feet shorter than } \\ \text { twice the shortest side. The third side is } 6 \text { feet longer than the shortest side. The perimeter is } 59 \text { feet. How long is each side? }\end{array} \)

To solve the problem, we need to define the sides of the triangle based on the information given. Let's denote: - \( x \): the length of the shortest side (in feet). - The longest side is \( 2x - 7 \) (7 feet shorter than twice the shortest side). - The third side is \( x + 6 \) (6 feet longer than the shortest side). Now, we can set up the equation for the perimeter of the triangle: \[ x + (2x - 7) + (x + 6) = 59 \] Now, let's simplify this equation step by step: 1. Combine like terms: \[ x + 2x - 7 + x + 6 = 59 \] \[ 4x - 1 = 59 \] 2. Add 1 to both sides: \[ 4x = 60 \] 3. Divide by 4: \[ x = 15 \] Now that we have the length of the shortest side, we can find the lengths of the other two sides: - Longest side: \[ 2x - 7 = 2(15) - 7 = 30 - 7 = 23 \] - Third side: \[ x + 6 = 15 + 6 = 21 \] Thus, the lengths of the sides of the triangle are: - Shortest side: \( 15 \) feet - Longest side: \( 23 \) feet - Third side: \( 21 \) feet To summarize, the lengths of each side are: - Shortest side: \( 15 \) feet - Longest side: \( 23 \) feet - Third side: \( 21 \) feet