Find the next shape in a repeating pattern

Learning Objectives Identify the core repeating unit in a geometric pattern of congruent triangles. Describe the specific sequence of rigid transformations (translation, rotation, reflection) that generates a pattern. Use congruence postulates (SSS, SAS, ASA) to justify that shapes within a pattern are congruent. Apply algebraic rules for transformations on a coordinate plane to predict the vertices of the next triangle in a sequence. Determine the position and orientation of the nth triangle in a repeating pattern. Justify their prediction for the next shape using the precise mathematical language of geometry and transformations. Ever noticed the beautiful, repeating patterns on a tile floor or in a kaleidoscope? 💠 How do artists and designers create those perfect, endless...

TermDefinitionExample Congruent TrianglesTriangles that have the exact same size and shape. All pairs of corresponding sides and corresponding angles are equal.If ΔABC ≅ ΔXYZ, it means AB = XY, BC = YZ, AC = XZ, and ∠A = ∠X, ∠B = ∠Y, ∠C = ∠Z. Rigid Transformation (Isometry)A transformation of the plane that preserves distance and angle measure. The original figure (pre-image) and the transformed figure (image) are congruent.Translations, rotations, and reflections are the three main rigid transformations. TranslationA rigid transformation that 'slides' every point of a figure the same distance in the same direction along a vector.Translating ΔABC with vertices A(1,2), B(3,1), C(2,4) by the vector <4, -1> results in ΔA'B'C' with vertices A'(5,1), B'...

Translation Rule T_{a,b}(x, y) = (x + a, y + b) Use this to find the coordinates of a point after sliding it 'a' units horizontally and 'b' units vertically. If 'a' is negative, the slide is to the left; if 'b' is negative, the slide is down. Rotation Rules (about the origin) 90° counter-clockwise: R_{90°}(x, y) = (-y, x) \\ 180°: R_{180°}(x, y) = (-x, -y) \\ 270° counter-clockwise: R_{270°}(x, y) = (y, -x) Use these rules to find the coordinates of a point after rotating it around the origin (0,0). For clockwise rotations, use the equivalent counter-clockwise rotation (e.g., 90° clockwise is 270° counter-clockwise). Reflection Rules Across x-axis: r_{x-axis}(x, y) = (x, -y) \\ Across y-axis: r_{y-axis}(x, y) = (-x, y) \\ Across...