AMC 8 2026 Problem 13
AMC8 2026 年第 13 题
坐标几何★★★☆☆
The figure below shows a tiling of 1×1 unit squares. Each row of unit squares is shifted horizontally by half a unit relative to the row above it. A shaded square is drawn on top of the tiling. Each vertex of the shaded square is a vertex of one of the unit squares. In square units, what is the area of the shaded square?
下图展示了一种由 1×1 单位正方形组成的铺砖图案。每一行相对于上一行水平偏移半个单位。图中还有一个覆盖在铺砖上的阴影正方形，其每个顶点都是某个单位正方形的顶点。这个阴影正方形的面积是多少（单位：平方单位）？
- A.10
- B.
- C.
- D.11
- E.
正确答案 · Correct Answer
A
解析 · Solution
解法1
正方形的边是一个直角三角形的斜边，两直角边为 1 cm 和 3 cm。由勾股定理：边长 cm。面积 cm²。
构造一个 4×4 的正方形恰好外接倾斜的正方形（四个角相切）。大正方形面积 16。四个角的直角三角形直角边分别为 1 和 3，每个面积 1.5。减去：16 − 4×1.5 = 10。
Solution 1
The side of the square is the hypotenuse of a right triangle with legs 1 cm and 3 cm. By the Pythagorean theorem: side = cm. Area = cm².
Construct a 4×4 square that circumscribes the tilted square, touching at all four corners. The 4×4 square has area 16. The four corner right triangles each have legs 1 and 3, area = (1×3)/2 = 1.5 each. Subtracting: 16 − 4×1.5 = 16 − 6 = 10.
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