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Hypotenuse of 45°-45°-90° Equals Hypotenuse of 30°-60°-90°

Hypotenuse of 45°-45°-90° Equals Hypotenuse of 30°-60°-90°
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2018-01-18 08:43
TrianglePolygonAngleTrigonometryGeometric Construction
①This GeoGebra construction visually demonstrates that the hypotenuse of a 45°-45°-90° right triangle can be made equal in length to the hypotenuse of a 30°-60°-90° right triangle, highlighting the relationship between the two special right triangles. ②Core mathematical concepts include the side-length ratios of special right triangles (1:1:√2 for 45-45-90 and 1:√3:2 for 30-60-90), the geometric equivalence of equal hypotenuses, and the trigonometric connections between angles and side lengths. ③Construction details: using GeoGebra commands such as Segment, Rotate, Angle, Ray, Intersect, and Polygon, the construction creates multiple triangles (poly1, poly2) and auxiliary lines (a, b, c, d), with rotated points (A', B', C') and angle labels (α, β, γ, δ, ε, ζ) plus text (text1) to annotate the relationships. ④This is suitable for middle school advanced students or high school learners studying special right triangles, and serves as an interactive classroom demonstration for comparing geometric constructions with equal hypotenuses.

GeoGebra Commands & Parameters

  1. Segment(D, C) → f
  2. Rotate(C, 90°, D) → B'
  3. Angle(C, D, B') → α
  4. Rotate(D, (-(45°)), C) → A'
  5. Angle(A', C, D) → β
  6. Ray(D, B') → g
  7. Ray(C, A') → h
  8. Intersect(g, h) → A
  9. Rotate(C, 60°, A) → B'_1
  10. Angle(C, A, B'_1) → γ
  11. Ray(A, B'_1) → i
  12. Rotate(A, (-(30°)), C) → C'
  13. Angle(C', C, A) → δ
  14. Ray(C, C') → j
  15. Intersect(i, j) → B
  16. Polygon(C, D, A) → poly1, c, b, a
  17. Polygon(A, B, C) → poly2, b_1, c_1, d
  18. Angle(A, B, C) → ε
  19. Angle(D, A, C) → ζ

AI Drawing Prompt Reference

Draw two right triangles sharing a common base segment, one being a 45-45-90 triangle and the other a 30-60-90 triangle, with equal hypotenuses. Start with a horizontal base. From its left endpoint, rotate a ray 90 degrees counterclockwise; from its right endpoint, rotate a ray 45 degrees clockwise; intersect these rays to form the apex of the first triangle, then connect the base endpoints to this apex to form the isosceles right triangle. Next, from this apex, rotate a ray 60 degrees counterclockwise toward the right side, and from the right base endpoint, rotate a ray 30 degrees clockwise; intersect these to form the apex of the second triangle, then connect the relevant points to form the 30-60-90 triangle. Label all acute angles and add a text note that the two hypotenuses are equal in length.
The prompt above was compiled from the original drawing commands and is for reference only.