A unified approach to function–geometry problems in the senior high school entrance examination
中考函数几何综合题的“多题一解”探析

摘要

中考函数与几何综合题是区分学生数形转化与迁移能力的关键题型。本文从“多题一解”视角，选取2024年苏州中考线段关系题、苏州中考面积最值题及湖北中考角相等题三类典型真题，系统剖析其解法共性。研究发现，三类题目虽设问不同——分别涉及线段位置判定、动态面积最大化及角度等量关系——但均可归结为“坐标表示→关系转化→方程（函数）求解→结果检验”的统一解题路径。其中，线段关系借助对称轴用坐标差表示长度；面积问题通过参数化动点坐标建立二次函数求最值；角度问题则通过作垂线构造直角三角形，将正切值相等转化为比例方程。该路径强调从图形条件到代数表达的可迁移转化策略。教学上，应引导学生在不同题型中识别共同结构，逐步实现从孤立解题到类题通解的能力跃升，为中考专题复习提供结构化思路。

Abstract

Function–geometry problems in the senior high school entrance examination are important for testing students’ ability to connect geometric figures with algebraic expressions and apply problem-solving methods to similar problems. This paper discusses three typical problems from the 2024 entrance examinations: a line-segment relation problem and an area maximum problem from Suzhou, and an equal-angle problem from Hubei. Although the three problems ask different questions, their solutions follow a similar four-step process: coordinate representation, relation transformation, equation or function solving, and result checking. In the line-segment problem, the axis of symmetry is used to express segment lengths by coordinate differences. In the area problem, the coordinates of a moving point are expressed with a parameter, and a quadratic function is then used to find the maximum area. In the angle problem, perpendicular lines are added to form right triangles, and equal tangent values are used to transform the angle relation into a proportional equation. These three methods show a common way of turning geometric conditions into algebraic relations. In teaching, students should be guided to identify the common structure behind different types of problems, so that they can gradually move from solving a single problem to mastering a general method for a class of problems. This can provide a clearer and more systematic approach to topic-based review for the senior high school entrance examination.