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Geometric Demonstration of the Square of a Difference

Geometric Demonstration of the Square of a Difference
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2023-03-19 01:32
TrianglePolygonVectorTransformationGeometric Construction
In collections geometriadinamica.es
① This GeoGebra example visually demonstrates the algebraic identity (a-b)² = a² - 2ab + b², known as the square of a difference. ② Core mathematical concepts include the perfect square formula, area decomposition, vector translation, right triangles, and median properties. ③ Construction highlights: using GeoGebra commands such as Polygon and Vector, the construction creates multiple polygons (e.g., T1, T2, T5) and translation vectors (u, v, w, z). Points are translated (A→AA, C→CC, B→BBB) to form auxiliary shapes, while segments and medians (a, c, e, g, h) help build right triangles and area relationships. The large square is partitioned into two smaller squares and two rectangles, clearly illustrating the formula. ④ Suitable for middle and high school students learning the connection between algebra and geometry, and for teachers to use in classroom demonstrations to reveal the geometric origin of algebraic identities.

GeoGebra Commands & Parameters

  1. Polygon(A, C, I, E) → R, a_2, c_1, i, e_1
  2. Vector(AA, CC) → u
  3. Vector(CC, AA) → v
  4. Vector(BBB, CCC) → w
  5. Vector(CCC, BBB) → z
  6. Polygon(A, B, F, D) → P, a_1, b_1, f, d
  7. Polygon(B, C, I, H) → Q, c, e_{2}, g, h
  8. Polygon(G, I, E, D) → S, e, j, k, l

AI Drawing Prompt Reference

Create a geometric demonstration of the square of a difference identity by decomposing a large square region into multiple parts: two smaller squares and two rectangles arranged so their shapes and positions reflect the formula (a−b)^2 = a^2 − 2ab + b^2. Start by drawing the main square partition using several polygons connected to chosen key points, ensuring some right triangles and medial/line segments are evident to guide the area relationships. Then apply vector translations to move key points to new locations, forming auxiliary translated contours and additional polygonal pieces. Finally, assemble all the polygons so the partitions visually match the intended area decomposition across the entire large square.
The prompt above was compiled from the original drawing commands and is for reference only.