Folding and Symmetry Line Investigation

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2018-04-25 02:37

Polygon Geometric Construction

This example demonstrates how to investigate the lines of symmetry of a shape through folding operations and determine the number of these lines. The core mathematical concepts involved include symmetry, folding transformation, and geometric construction. Construction highlights involve using commands and objects in GeoGebra such as polygon, segment, intersect, circle, point, line, circle sector, mirror, midpoint, line bisector, rotate, angle, translate, and textfield. Suitable for middle school and above students learning about symmetry and geometric transformations.

GeoGebra Commands & Parameters

Polygon(A, B, 4) → mnogokut1, f, g, h, i, C, D
Segment(B, D) → k
Segment(A, C) → j
Intersect(j, k) → E
Circle(E, B) → c
Point(c) → F
Line(F, E) → l
Intersect(c, l, 2) → G
CircleSector(E, G, F) → polukrug
Point(l) → H
Point(l) → I
Segment(H, I) → m
Polygon(J, J', 4) → mnogokut2, n, p, q, r, K, L
Segment(J, K) → s
Segment(J', L) → t
Intersect(s, t) → M
Line(M, l) → a
Point(a) → N
Point(a) → O
Segment(N, O) → b
Polygon(A, B, T_1) → mnogokut3, t_1, a_1, b_1
Mirror(A, l) → A'
Mirror(B, l) → B'
Mirror(T_1, l) → P
Polygon(A', B', P) → mnogokut3', p_1, a', b'
Polygon(J, J', T_2) → mnogokut3_1, t_2, j_1, j'
Midpoint(Segment[D, C]) → T_3
Polygon(A, B, T_3) → mnogokut4, t_3, a_2, b_2
Midpoint(Segment[L, K]) → T_4
Polygon(J, J', T_4) → mnogokut4_1, t_4, j_2, j'_1
Mirror(A, l) → A'_1
Mirror(B, l) → B'_1
Mirror(T_3, l) → Q
Polygon(A'_1, B'_1, Q) → mnogokut4', q_1, a'_1, b'_1
Polygon(A, B, 3) → mnogokut5, d, e, f_1, R
LineBisector(d) → g_1
LineBisector(e) → h_1
Intersect(g_1, h_1) → S
Circle(S, B) → c_1
Point(c_1) → T
Line(T, S) → i_1
Intersect(c_1, i_1, 2) → U
CircleSector(S, U, T) → d_1
Point(i_1) → V
Point(i_1) → W
Segment(V, W) → k_1
Mirror(mnogokut5, i_1) → mnogokut5'
Polygon(J, J', 3) → mnogokut6, l_1, m_1, r_1, Z
LineBisector(l_1) → s_1
LineBisector(m_1) → e_1
Intersect(s_1, e_1) → A_1
Line(A_1, k_1) → f_2
Point(f_2) → B_1
Point(f_2) → C_1
Segment(C_1, B_1) → g_2
Polygon(A, B, 4) → mnogokut7, h_2, i_2, k_2, l_2, D_1, E_1
Mirror(mnogokut7, l) → mnogokut7'
Polygon(J, J', 4) → mnogokut7_1, m_2, n_2, p_2, q_2, F_1, G_1
Line(A, I_1) → r_2
Circle(A, Segment[A, H_1]) → c_2
Intersect(c_2, r_2, 2) → J_1
Circle(J_1, Segment[A, H_1]) → d_2
Circle(H_1, Segment[A, H_1]) → e_2
Intersect(d_2, e_2, 1) → K_1
Polygon(A, H_1, K_1, J_1) → mnogokut8, a_3, h_3, k_3, j_3
Segment(A, K_1) → s_2
Segment(H_1, J_1) → f_3
Intersect(s_2, f_3) → L_1
Circle(L_1, K_1) → c_3
Point(c_3) → M_1
Line(M_1, L_1) → g_3
Intersect(c_3, g_3, 2) → N_1
CircleSector(L_1, N_1, M_1) → d_3
Mirror(A, g_3) → A'_2
Mirror(H_1, g_3) → O_1
Mirror(K_1, g_3) → P_1
Mirror(J_1, g_3) → Q_1
Polygon(A'_2, O_1, P_1, Q_1) → mnogokut8', a'_2, o_1, p_3, q_3
Point(g_3) → R_1
Point(g_3) → S_1
Segment(S_1, R_1) → i_3
Line(J, r_2) → l_3
Circle(J, Segment[J, U_1]) → e_3
Intersect(e_3, l_3, 2) → V_1
Circle(V_1, Segment[J, U_1]) → r_3
Circle(U_1, Segment[J, U_1]) → s_3
Intersect(r_3, s_3, 1) → W_1
Polygon(J, U_1, W_1, V_1) → mnogokut9, j_4, u_1, w_1, v_1
Segment(J, W_1) → m_3
Segment(U_1, V_1) → n_3
Intersect(m_3, n_3) → Z_1
Line(Z_1, g_3) → b_3
Point(b_3) → A_2
Point(b_3) → B_2
Segment(B_2, A_2) → f_4
Circle(A, Segment[B, C_2]) → c_4
Circle(C_2, Segment[A, B]) → d_4
Intersect(c_4, d_4, 1) → D_2
Polygon(A, B, C_2, D_2) → mnogokut10, a_4, b_4, c_5, d_5
Segment(A, C_2) → g_4
Segment(B, D_2) → h_4
Intersect(g_4, h_4) → E_2
Circle(E_2, B) → e_4
Point(e_4) → F_2
Line(F_2, E_2) → i_4
Intersect(e_4, i_4, 2) → G_2
CircleSector(E_2, G_2, F_2) → k_4
Mirror(A, i_4) → A'_3
Mirror(B, i_4) → B'_2
Mirror(C_2, i_4) → H_2
Mirror(D_2, i_4) → I_2
Polygon(A'_3, B'_2, H_2, I_2) → mnogokut10', a'_3, b'_2, h_5, i_5
Point(i_4) → J_2
Point(i_4) → K_2
Circle(J, Segment[J', L_2]) → p_4
Circle(L_2, Segment[J, J']) → q_4
Intersect(p_4, q_4, 1) → M_2
Polygon(J, J', L_2, M_2) → mnogokut10_1, j_5, j'_2, l_4, m_4
Segment(J, L_2) → n_4
Segment(J', M_2) → r_4
Intersect(n_4, r_4) → N_2
Line(N_2, i_4) → s_4
Point(s_4) → O_2
Point(s_4) → P_2
Segment(K_2, J_2) → f_5
Segment(P_2, O_2) → g_5
Polygon(A, B, R_2, Q_2) → mnogokut11, a_5, b_5, r_5, q_5
Mirror(A, l) → A'_4
Mirror(B, l) → B'_3
Mirror(R_2, l) → S_2
Mirror(Q_2, l) → U_2
Polygon(A'_4, B'_3, S_2, U_2) → mnogokut11', a'_4, b'_3, s_5, u_2
Polygon(J, J', W_2, V_2) → mnogokut12, j_6, j'_3, w_2, v_2
Circle(A, 3) → e_5
Intersect(e_5, r_2, 2) → Z_2
Circle(Z_2, Segment[A, B]) → k_5
Circle(B, Segment[A, Z_2]) → p_5
Intersect(k_5, p_5, 1) → A_3
Polygon(A, B, A_3, Z_2) → mnogokut13, a_6, b_6, a_7, z_2
Segment(A, A_3) → l_5
Segment(B, Z_2) → m_5
Intersect(l_5, m_5) → B_3
Circle(B_3, A_3) → t_5
Point(t_5) → C_3
Line(C_3, B_3) → n_5
Intersect(t_5, n_5, 2) → D_3
Line(J, r_2) → f_6
CircleSector(B_3, D_3, C_3) → c_6
Mirror(A, n_5) → A'_5
Mirror(B, n_5) → B'_4
Mirror(A_3, n_5) → E_3
Mirror(Z_2, n_5) → F_3
Polygon(A'_5, B'_4, E_3, F_3) → mnogokut13', a'_5, b'_4, e_6, f_7
Point(n_5) → G_3
Point(n_5) → H_3
Circle(J, 3) → d_6
Intersect(d_6, f_6, 2) → I_3
Circle(I_3, Segment[J, J']) → g_6
Circle(J', Segment[J, I_3]) → h_6
Intersect(g_6, h_6, 1) → J_3
Polygon(J, J', J_3, I_3) → mnogokut13_1, j_7, j'_4, j_8, i_6
Segment(J, J_3) → k_6
Segment(J', I_3) → l_6
Intersect(k_6, l_6) → K_3
Line(K_3, n_5) → m_6
Point(m_6) → L_3
Point(m_6) → M_3
Segment(H_3, G_3) → n_6
Segment(M_3, L_3) → p_6
Line(E, yAxis) → q_6
Intersect(c, q_6, 1) → N_3
Rotate(N_3, 36°, E) → O_3
Angle(N_3, E, O_3) → α
Rotate(N_3, (-(36°)), E) → P_3
Angle(P_3, E, N_3) → β
Circle(O_3, Segment[P_3, O_3]) → r_6
Intersect(c, r_6, 1) → Q_3
Circle(Q_3, Segment[O_3, Q_3]) → s_6
Intersect(c, s_6, 1) → R_3
Circle(R_3, Segment[Q_3, R_3]) → t_6
Intersect(c, t_6, 1) → S_3
Polygon(P_3, O_3, Q_3, R_3, S_3) → mnogokut14, p_7, o_3, q_7, r_7, s_7
Mirror(P_3, l) → U_3
Mirror(O_3, l) → V_3
Mirror(Q_3, l) → W_3
Mirror(R_3, l) → Z_3
Mirror(S_3, l) → A_4
Polygon(U_3, V_3, W_3, Z_3, A_4) → mnogokut14', u_3, v_3, w_3, z_3, a_8
Translate(P_3, Vector[A, J]) → B_4
Translate(O_3, Vector[A, J]) → C_4
Translate(Q_3, Vector[A, J]) → D_4
Translate(R_3, Vector[A, J]) → E_4
Translate(S_3, Vector[A, J]) → F_4
Polygon(B_4, C_4, D_4, E_4, F_4) → mnogokut14'_1, b_7, c_7, d_7, e_7, f_8
Rotate(N_3, 30°, E) → G_4
Angle(N_3, E, G_4) → γ
Rotate(N_3, (-(30°)), E) → H_4
Angle(H_4, E, N_3) → δ
Circle(G_4, Segment[H_4, G_4]) → g_7
Intersect(c, g_7, 1) → I_4
Circle(I_4, Segment[G_4, I_4]) → h_7
Intersect(c, h_7, 1) → J_4
Circle(J_4, Segment[I_4, J_4]) → k_7
Intersect(c, k_7, 1) → K_4
Circle(K_4, Segment[J_4, K_4]) → t_7
Intersect(c, t_7, 1) → L_4
Polygon(H_4, G_4, I_4, J_4, K_4, L_4) → mnogokut15, h_8, g_8, i_7, j_9, k_8, l_7
Mirror(H_4, l) → M_4
Mirror(G_4, l) → N_4
Mirror(I_4, l) → O_4
Mirror(J_4, l) → P_4
Mirror(K_4, l) → Q_4
Mirror(L_4, l) → R_4
Polygon(M_4, N_4, O_4, P_4, Q_4, R_4) → mnogokut15', m_7, n_7, o_4, p_8, q_8, r_8
Translate(H_4, Vector[A, J]) → S_4
Translate(G_4, Vector[A, J]) → U_4
Translate(I_4, Vector[A, J]) → V_4
Translate(J_4, Vector[A, J]) → W_4
Translate(K_4, Vector[A, J]) → Z_4
Translate(L_4, Vector[A, J]) → A_5
Polygon(S_4, U_4, V_4, W_4, Z_4, A_5) → mnogokut15'_1, s_8, u_4, v_4, w_4, z_4, a_9
Polygon(B_5, C_5, D_5, L_5, M_5, E_5, I_5, K_5, H_5, F_5, J_5, G_5) → mnogokut16, b_8, g_9, d_8, l_8, c_8, m_8, e_8, i_8, k_9, h_9, f_9, j_{10}
Mirror(B_5, l) → N_5
Mirror(C_5, l) → O_5
Mirror(M_5, l) → P_5
Mirror(E_5, l) → Q_5
Mirror(I_5, l) → R_5
Mirror(K_5, l) → S_5
Mirror(H_5, l) → T_5
Mirror(F_5, l) → U_5
Mirror(J_5, l) → V_5
Mirror(G_5, l) → W_5
Mirror(D_5, l) → Z_5
Mirror(L_5, l) → A_6
Polygon(N_5, O_5, Z_5, A_6, P_5, Q_5, R_5, S_5, T_5, U_5, V_5, W_5) → mnogokut16', n_8, w_5, z_5, a_{10}, o_5, p_9, q_9, r_9, s_9, t_8, u_5, v_5
Translate(B_5, Vector[A, J]) → B_6
Translate(C_5, Vector[A, J]) → C_6
Translate(M_5, Vector[A, J]) → D_6
Translate(E_5, Vector[A, J]) → E_6
Translate(I_5, Vector[A, J]) → F_6
Translate(K_5, Vector[A, J]) → G_6
Translate(H_5, Vector[A, J]) → H_6
Translate(F_5, Vector[A, J]) → I_6
Translate(J_5, Vector[A, J]) → J_6
Translate(G_5, Vector[A, J]) → K_6
Translate(D_5, Vector[A, J]) → L_6
Translate(L_5, Vector[A, J]) → M_6
Polygon(B_6, C_6, L_6, M_6, D_6, E_6, F_6, G_6, H_6, I_6, J_6, K_6) → mnogokut16'_1, b_9, k_{10}, l_9, m_9, c_9, d_9, e_9, f_{10}, g_{10}, h_{10}, i_9, j_{11}
Polygon(A, P_6, N_6, D, S_6, U_6, Q_6, B, C, O_6, T_6, R_6) → mnogokut17, a_{11}, r_{10}, n_9, d_{10}, p_{10}, s_{10}, u_6, q_{10}, b_{10}, c_{10}, o_6, t_9
Mirror(A, l) → A'_6
Mirror(P_6, l) → V_6
Mirror(S_6, l) → W_6
Mirror(U_6, l) → Z_6
Mirror(Q_6, l) → A_7
Mirror(B, l) → B'_5
Mirror(C, l) → C'
Mirror(O_6, l) → B_7
Mirror(T_6, l) → C_7
Mirror(R_6, l) → D_7
Mirror(N_6, l) → E_7
Mirror(D, l) → D'
Polygon(A'_6, V_6, E_7, D', W_6, Z_6, A_7, B'_5, C', B_7, C_7, D_7) → mnogokut17', a'_6, d_{11}, e_{10}, d', v_6, w_6, z_6, a_{12}, b'_5, c', b_{11}, c_{11}
Translate(A, Vector[A, J]) → A'_7
Translate(P_6, Vector[A, J]) → F_7
Translate(S_6, Vector[A, J]) → G_7
Translate(U_6, Vector[A, J]) → H_7
Translate(Q_6, Vector[A, J]) → I_7
Translate(B, Vector[A, J]) → B'_6
Translate(C, Vector[A, J]) → C'_1
Translate(O_6, Vector[A, J]) → J_7
Translate(T_6, Vector[A, J]) → K_7
Translate(R_6, Vector[A, J]) → L_7
Translate(N_6, Vector[A, J]) → M_7
Translate(D, Vector[A, J]) → D'_1
Polygon(A'_7, F_7, M_7, D'_1, G_7, H_7, I_7, B'_6, C'_1, J_7, K_7, L_7) → mnogokut17'_1, a'_7, l_{10}, m_{10}, d'_1, f_{11}, g_{11}, h_{11}, i_{10}, b'_6, c'_1, j_{12}, k_{11}
Polygon(R_7, S_7, Q_7, O_7, C, D, N_7, P_7) → mnogokut18, r_{11}, s_{11}, q_{11}, o_7, c_{12}, d_{12}, n_{10}, p_{11}
Mirror(R_7, l) → T_7
Mirror(S_7, l) → U_7
Mirror(Q_7, l) → V_7
Mirror(O_7, l) → W_7
Mirror(C, l) → C'_2
Mirror(D, l) → D'_2
Mirror(N_7, l) → Z_7
Mirror(P_7, l) → A_8
Polygon(T_7, U_7, V_7, W_7, C'_2, D'_2, Z_7, A_8) → mnogokut18', t_{10}, u_7, v_7, w_7, c'_2, d'_2, z_7, a_{13}
Translate(R_7, Vector[A, J]) → B_8
Translate(S_7, Vector[A, J]) → C_8
Translate(Q_7, Vector[A, J]) → D_8
Translate(O_7, Vector[A, J]) → E_8
Translate(C, Vector[A, J]) → C'_3
Translate(D, Vector[A, J]) → D'_3
Translate(N_7, Vector[A, J]) → F_8
Translate(P_7, Vector[A, J]) → G_8
Polygon(B_8, C_8, D_8, E_8, C'_3, D'_3, F_8, G_8) → mnogokut18'_1, b_{12}, c_{13}, d_{13}, e_{11}, c'_3, d'_3, f_{12}, g_{12}
Polygon(D, H_8, A, O_8, M_8, I_8, C, N_8, B, K_8, L_8, J_8) → mnogokut19, d_{14}, j_{13}, a_{14}, o_8, h_{12}, m_{11}, i_{11}, c_{14}, n_{11}, b_{13}, k_{12}, l_{11}
Mirror(D, l) → D'_4
Mirror(H_8, l) → P_8
Mirror(M_8, l) → Q_8
Mirror(I_8, l) → R_8
Mirror(C, l) → C'_4
Mirror(N_8, l) → S_8
Mirror(B, l) → B'_7
Mirror(K_8, l) → T_8
Mirror(L_8, l) → U_8
Mirror(J_8, l) → V_8
Mirror(A, l) → A'_8
Mirror(O_8, l) → W_8
Polygon(D'_4, P_8, A'_8, W_8, Q_8, R_8, C'_4, S_8, B'_7, T_8, U_8, V_8) → mnogokut19', d'_4, v_8, a'_8, w_8, p_{12}, q_{12}, r_{12}, c'_4, s_{12}, b'_7, t_{11}, u_8
Translate(D, Vector[A, J]) → D'_5
Translate(H_8, Vector[A, J]) → Z_8
Translate(M_8, Vector[A, J]) → A_9
Translate(I_8, Vector[A, J]) → B_9
Translate(C, Vector[A, J]) → C'_5
Translate(N_8, Vector[A, J]) → C_9
Translate(B, Vector[A, J]) → B'_8
Translate(K_8, Vector[A, J]) → D_9
Translate(L_8, Vector[A, J]) → E_9
Translate(J_8, Vector[A, J]) → F_9
Translate(A, Vector[A, J]) → A'_9
Translate(O_8, Vector[A, J]) → G_9
Polygon(D'_5, Z_8, A'_9, G_9, A_9, B_9, C'_5, C_9, B'_8, D_9, E_9, F_9) → mnogokut19'_1, d'_5, f_{13}, a'_9, g_{13}, z_8, a_{15}, b_{14}, c'_5, c_{15}, b'_8, d_{15}, e_{12}
Textfield(brOsiUpisano) → txtPolje1

AI Drawing Prompt Reference

Draw a quadrilateral, find the intersection point of its diagonals, and draw a circle centered at this point. Mark a point on the circle, draw a line through this point and the intersection point, and find another intersection point on the circle. Draw a sector with the intersection point and the center. Mirror both the quadrilateral and the sector over the line through the intersection points to create two new shapes. Then, draw a triangle, find the intersection point of the perpendicular bisectors of its two sides, draw a circle centered at this point, mark two points on the circle, and draw a line through them. Mirror the triangle over the line through these points.

The prompt above was compiled from the original drawing commands and is for reference only.