求第10题答案---**Question 10** **Question Stem:** 如图,将等边△ABC沿EF所在直线折叠,使点C落在AB的中点D处,P是EF上一个动点,连接PA,PD,若AB=8,则△PAD周长的最小值为______. **English Translation of Question Stem:** As shown in the figure, fold the equilateral triangle ABC along the line EF, such that point C lands on the midpoint D of AB. P is a moving point on EF. Connect PA and PD. If AB=8, then the minimum value of the perimeter of △PAD is ______. **Options:** [No options provided in the image] **Other Relevant Text:** 第 10 题图 (Figure for Question 10) 41 (Likely a page number or problem set number, not directly part of the question content itself) **Geometric Figure Description:** * **Type:** Geometric figure showing a triangle and a folding operation. * **Main Elements:** * An equilateral triangle ABC. * Point D is the midpoint of side AB. * A line segment EF is shown, where E is on AB and F is on AC. * The triangle is folded along the line containing EF, causing point C to coincide with point D. * This folding implies that EF is the perpendicular bisector of the line segment CD. * Point P is shown on the line segment EF. * Line segments PA and PD are drawn, forming triangle PAD. * The original position of triangle ABC is indicated by dashed lines for sides AC and BC. * Point C is shown outside the folded region, and D is shown inside the triangle, on AB. **Mathematical Formulas/Equations:** * AB = 8 * △ABC is equilateral. * D is the midpoint of AB. * Perimeter of △PAD = PA + PD + AD. **Inferred Information from Folding:** Since C is folded onto D along EF, for any point P on EF, the distance from P to C is equal to the distance from P to D (PC = PD). **Solution Approach (Inferred):** To minimize the perimeter of △PAD, which is PA + PD + AD, we need to minimize PA + PD since AD is a fixed length. Using the property PC = PD, minimizing PA + PD is equivalent to minimizing PA + PC. The sum PA + PC is minimized when P lies on the straight line segment AC. Since P must also lie on the line EF, the minimum value of PA + PC occurs at the intersection of the line AC and the line EF, provided this intersection point is on the segment EF. By the reflection principle (or simply PC=PD), the path PA+PD is minimized when A, P, and the original position of C are collinear. Therefore, the minimum value of PA + PD is the length of the segment AC. Since △ABC is equilateral with AB=8, AC=8. AD is half of AB, so AD = 8/2 = 4. The minimum perimeter of △PAD is the minimum of (PA + PD) + AD = AC + AD = 8 + 4 = 12. **Answer (Calculated based on the inferred solution):** 12