按照图片中解析的思路，生成本题的视频讲解---**Extraction Content:** **Question Stem:** 如图，在Rt△ABC中，∠BAC = 90°, AB = 3, BC = 5, 点P为BC边上任意一点，连接PA, 以PA, PC为邻边作平行四边形PAQC，连接PQ, 则PQ长度的最小值为（ ） **Options:** [Options are not visible in the provided image] The text "故选：C." at the end of the solution indicates that options were present and option C was the correct answer. **Chart/Diagram Description:** Type: Geometric figure consisting of a right-angled triangle and a parallelogram. Main Elements: * Triangle ABC: A right-angled triangle with angle BAC = 90 degrees. Vertices are labeled A, B, C. Side lengths are given as AB = 3 and BC = 5. * Point P: A point located on the line segment BC. Its position on BC is variable ("任意一点"). * Quadrilateral APQC: A parallelogram formed with PA and PC as adjacent sides. Its vertices are labeled A, P, Q, C. * Lines: The sides of the triangle AB, AC, BC are shown. Segments PA and PC connect vertices A and P, and P and C respectively. Segments AQ and CQ are the other two sides of the parallelogram. The diagonals of the parallelogram, AC and PQ, are also drawn. * Intersection: The diagonals AC and PQ intersect at a point (labeled O in the accompanying solution). * Labels: Vertices A, B, C, P, Q are labeled. * Relative Position: Point P lies on the line segment BC. **Other Relevant Text (提示与解析 - Hints and Solution):** 提示与解析 标准解析 解：∵∠BAC = 90°, AB = 3, BC = 5, ∴AC = √BC² - AB² = √5² - 3² = √25 - 9 = √16 = 4, ∵四边形APCQ是平行四边形， ∴PO = QO, CO = AO, (This implies O is the intersection of diagonals AC and PQ) ∴PQ最短也就是PO最短, ∵过O作BC的垂线OP', (Here, P' is used to denote the foot of the perpendicular from O to BC) ∴∠ACB = ∠P'CO, ∠CP'O = ∠CAB = 90°, (∠P'CO is the same angle as ∠ACB) ∴△CAB ~ △CP'O, ∴CO / BC = OP' / AB, ∴2 / 5 = OP' / 3, (This step implies CO = AC / 2. Since O is the intersection of diagonals of parallelogram APQC, and AC is a diagonal, O is the midpoint of AC. Therefore CO = AO = AC/2. Since AC=4, CO=2) ∴OP' = 6 / 5, ∴则PQ的最小值为2OP' = 2 * (6 / 5) = 12 / 5 = 2.4, (Since O is the midpoint of PQ, PQ = 2 * PO. The shortest distance from O to the line BC is the perpendicular distance OP'. The shortest length of PO occurs when P is the foot of the perpendicular from O to BC, i.e., P coincides with P'. Therefore, the minimum length of PO is OP', and the minimum length of PQ is 2 * OP') 故选：C.