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Exploring Similar Right Triangles by Tim Brzezinski

Exploring Similar Right Triangles by Tim Brzezinski
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2018-03-21 04:42
TriangleAngleTransformationGeometric ConstructionSlider Animation
In collections Similarity Theorems!
This GeoGebra applet allows students to explore the similarity of right triangles by drawing the altitude to the hypotenuse. Key mathematical concepts include similar triangles, geometric mean, and triangle properties. Using commands like Segment, LineBisector, CircleArc, students can dynamically explore the properties of right triangles. Suitable for middle school and above students learning about geometric transformations and similar triangles.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. LineBisector(a) → s_3
  3. Midpoint(a) → C_1
  4. Semicircle(A, B) → d
  5. Intersect(d, s_3, 1) → J_2
  6. CircleArc(C_1, B, J_2) → t_3
  7. Point(t_3) → C
  8. Segment(C, B) → f
  9. Segment(A, C) → e
  10. Min(a, f) → m
  11. Min(a, e) → l
  12. Min(l, m) → n
  13. OrthogonalLine(C_1, a) → b
  14. Circle(C_1, A) → c
  15. OrthogonalLine(C, a) → g
  16. Intersect(g, a) → E
  17. Segment(C, E) → h
  18. Angle(B, E, C) → α
  19. Angle(A, C, B) → β
  20. Polygon(A, E, C) → poly1, d_1, a_1, e_1
  21. Polygon(E, B, C) → poly2, d_2, e_2, b_1
  22. Circle(A, (0.2 * n)) → k
  23. Circle(C, (0.2 * n)) → p
  24. Circle(B, (0.2 * n)) → q
  25. Intersect(p, e_1, 1) → F
  26. Intersect(a_1, b_1) → G
  27. Intersect(a_1, b_1) → H
  28. Intersect(p, h, 1) → I
  29. Intersect(p, f, 1) → J
  30. Intersect(q, a, 1) → K
  31. Intersect(q, f, 1) → L
  32. Intersect(k, a, 1) → M
  33. Intersect(k, e, 1) → N
  34. CircleSector(A, M, N) → r
  35. Dilate(C, If[-2 < i < -1, -1 - i, 0], E) → D'
  36. Segment(C, D') → j
  37. CircleSector(B, L, K) → t
  38. Angle(C, E, A) → γ
  39. Dilate(A, If[-0.5 < i ≤ 0.5, 0.5 - i, 0], C) → A'
  40. Vector(A, A') → u
  41. Translate(r, u) → r'
  42. CircleArc(A, M, N) → f_1
  43. Dilate(B, If[1.5 < i ≤ 2.5, 2.5 - i, 0], C) → B'
  44. Vector(B, B') → v
  45. Translate(t, v) → t'
  46. CircleArc(B, L, K) → g_1
  47. Polygon(A, B, C) → poly3, d_3, a_2, b_2
  48. Rotate(r', If[0.5 < i ≤ 1.5, ((-(i - 0.5)) * π / 2), -1.5707963267948966], C) → r''
  49. Translate(t, v) → t'_1
  50. Rotate(t'_1, If[2.5 < i ≤ 3.5, (((i - 2.5) * 90))°, 90°], C) → s
  51. Mirror(E, f) → E'
  52. Mirror(E, e) → E'_1
  53. Dilate(E, If[4 < i ≤ 5, 5 - i, 0], E'_1) → E'_2
  54. Segment(C, E'_2) → h_1
  55. Segment(E'_2, A) → i_1
  56. Polygon(A, C, E'_2) → t1, e'_2, a_3, c_1
  57. Segment(A, C) → j_1
  58. Angle(A, E'_1, C) → δ
  59. Dilate(E, If[6 < i ≤ 7, 7 - i, 0], E') → E'_3
  60. Segment(C, E'_3) → k_1
  61. Segment(E'_3, B) → l_1
  62. Segment(C, B) → m_1
  63. Polygon(B, E'_3, C) → t2, c_2, b_3, e'_3
  64. Angle(C, E', B) → ε
  65. Angle(E, A, C) → ζ
  66. Centroid(t1) → D
  67. Centroid(t2) → O
  68. Centroid(poly3) → P
  69. Rotate(s, If[7 < i ≤ 8, ((-(i - 7)) * (π / 2 - ζ)), ζ - π / 2], C) → s'
  70. Rotate(r'', If[8 < i ≤ 9, ((i - 8) * ζ), ζ], C) → r'''
  71. Dilate(D, If[10 < i ≤ 11, 11 - i, 0], P) → D'_1
  72. Vector(D, D'_1) → w
  73. Translate(A, w) → A'_1
  74. Translate(C, w) → C'
  75. Translate(E'_2, w) → Q
  76. Polygon(A'_1, C', Q) → t1', q_1, a'_1, c'
  77. Translate(C, w) → C'_1
  78. Translate(E'_2, w) → R
  79. Segment(C'_1, R) → n_1
  80. Translate(E'_2, w) → S
  81. Translate(A, w) → A'_2
  82. Segment(S, A'_2) → p_1
  83. Translate(A, w) → A'_3
  84. Translate(C, w) → C'_2
  85. Segment(A'_3, C'_2) → r_1
  86. Translate(s', w) → s''
  87. Rotate(A'_1, If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → T
  88. Rotate(C', If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → C''
  89. Rotate(Q, If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → Q'
  90. Polygon(T, C'', Q') → t1'', q', t_1, c''
  91. Rotate(C'_1, If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → U
  92. Rotate(R, If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → R'
  93. Segment(U, R') → s_1
  94. Rotate(S, If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → S'
  95. Rotate(A'_2, If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → V
  96. Segment(S', V) → f_2
  97. Rotate(A'_3, If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → W
  98. Rotate(C'_2, If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → Z
  99. Segment(W, Z) → g_2
  100. Rotate(s'', If[10 < i ≤ 11, ((-(i - 10)) * ζ), (-ζ)], D'_1) → s'''
  101. Angle(T, S', U) → η
  102. Dilate(O, If[11 < i ≤ 12, 12 - i, 0], P) → O'
  103. Vector(O, O') → u_1
  104. Translate(B, u_1) → B'_1
  105. Translate(E'_3, u_1) → A_1
  106. Translate(C, u_1) → C'_3
  107. Polygon(B'_1, A_1, C'_3) → t2', c'_3, b'_1, a_4
  108. Translate(C, u_1) → C'_4
  109. Translate(E'_3, u_1) → D_1
  110. Segment(C'_4, D_1) → h_2
  111. Translate(E'_3, u_1) → E_1
  112. Translate(B, u_1) → B'_2
  113. Segment(E_1, B'_2) → i_2
  114. Translate(C, u_1) → C'_5
  115. Translate(B, u_1) → B'_3
  116. Segment(C'_5, B'_3) → j_2
  117. Translate(r''', u_1) → k_2
  118. Rotate(B'_1, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → B_1
  119. Rotate(A_1, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → F_1
  120. Rotate(C'_3, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → G_1
  121. Polygon(B_1, F_1, G_1) → t2'', g_3, b_4, f_3
  122. Rotate(C'_4, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → H_1
  123. Rotate(D_1, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → I_1
  124. Segment(H_1, I_1) → l_2
  125. Rotate(E_1, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → J_1
  126. Rotate(B'_2, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → K_1
  127. Segment(J_1, K_1) → m_2
  128. Rotate(C'_5, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → L_1
  129. Rotate(B'_3, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → M_1
  130. Segment(L_1, M_1) → n_2
  131. Rotate(k_2, If[11 < i ≤ 12, ((i - 11) * (π / 2 - ζ)), π / 2 - ζ], O') → p_2
  132. Angle(G_1, J_1, B_1) → θ
  133. Dilate(T, If[13 < i ≤ 14, 14 - i, 0], A) → T'
  134. Vector(T, T') → v_1
  135. Translate(T, v_1) → T'_1
  136. Translate(C'', v_1) → C'''
  137. Translate(Q', v_1) → Q''
  138. Polygon(T'_1, C''', Q'') → t1''', q'', t'_2, c'''
  139. Translate(S', v_1) → S''
  140. Translate(V, v_1) → V'
  141. Segment(S'', V') → q_2
  142. Translate(W, v_1) → W'
  143. Translate(Z, v_1) → Z'
  144. Segment(W', Z') → r_2
  145. Translate(U, v_1) → U'
  146. Translate(R', v_1) → R''
  147. Segment(U', R'') → s_2
  148. Translate(s''', v_1) → t_2
  149. Angle(Z', R'', V') → ι
  150. Dilate(G_1, If[13 < i ≤ 14, 14 - i, 0], A) → N_1
  151. Vector(G_1, N_1) → w_1
  152. Translate(B_1, w_1) → O_1
  153. Translate(F_1, w_1) → P_1
  154. Translate(G_1, w_1) → Q_1
  155. Polygon(O_1, P_1, Q_1) → t2''', q_3, o_1, p_3
  156. Translate(p_2, w_1) → c_3
  157. Translate(H_1, w_1) → R_1
  158. Translate(I_1, w_1) → S_1
  159. Segment(R_1, S_1) → h_3
  160. Translate(J_1, w_1) → T_1
  161. Translate(K_1, w_1) → U_1
  162. Segment(T_1, U_1) → i_3
  163. Translate(L_1, w_1) → V_1
  164. Translate(M_1, w_1) → W_1
  165. Segment(V_1, W_1) → j_3
  166. Angle(R_1, T_1, O_1) → κ
  167. Dilate(Z', If[15 < i ≤ 16, 16 - i, 0], B) → Z''
  168. Vector(Z', Z'') → u_2
  169. Translate(T'_1, u_2) → Z_1
  170. Translate(C''', u_2) → A_2
  171. Translate(Q'', u_2) → Q'''
  172. Polygon(Z_1, A_2, Q''') → t3, q''', z_1, a_5
  173. Translate(t_2, u_2) → e_3
  174. Translate(U', u_2) → U''
  175. Translate(R'', u_2) → R'''
  176. Segment(U'', R''') → k_3
  177. Translate(S'', u_2) → S'''
  178. Translate(V', u_2) → V''
  179. Segment(S''', V'') → l_3
  180. Angle(V'', S''', U'') → λ
  181. Dilate(O_1, If[15 < i ≤ 16, 16 - i, 0], B) → B_2
  182. Vector(O_1, B_2) → v_2
  183. Translate(O_1, v_2) → C_2
  184. Translate(P_1, v_2) → D_2
  185. Translate(Q_1, v_2) → E_2
  186. Polygon(C_2, D_2, E_2) → t4, e_4, c_4, d_4
  187. Translate(c_3, v_2) → r_3
  188. Translate(R_1, v_2) → F_2
  189. Translate(S_1, v_2) → G_2
  190. Segment(F_2, G_2) → m_3
  191. Translate(T_1, v_2) → H_2
  192. Translate(U_1, v_2) → I_2
  193. Segment(H_2, I_2) → n_3
  194. Angle(F_2, H_2, B_2) → μ

AI Drawing Prompt Reference

Draw a right triangle with a horizontal hypotenuse and its right-angle vertex above it. Drop a perpendicular from the right-angle vertex to the hypotenuse, dividing it into two segments. Construct a semicircle with the hypotenuse as diameter, placing the right-angle vertex on the semicircle. Draw a perpendicular at the midpoint of the hypotenuse, intersecting the semicircle, and use this to create an arc. Place small circles at the three vertices, and construct sectors at their intersections with triangle sides and the perpendicular. Add a slider to move the right-angle vertex along the arc, while the three small sectors translate or rotate accordingly, illustrating the dynamic similarity of the right triangles.
The prompt above was compiled from the original drawing commands and is for reference only.