Congruence transformations: mixed review (Tutorial Only)

Learning Objectives Identify the specific translation, reflection, or rotation that maps a pre-image to its image. Describe a sequence of two or more congruence transformations that maps one figure onto another. Apply algebraic rules to determine the coordinates of an image after a sequence of transformations. Write a coordinate rule for a given composition of congruence transformations. Prove two figures are congruent by defining a sequence of rigid motions that maps one onto the other. Differentiate between rigid motions (isometries) and non-rigid motions (like dilations). Have you ever wondered how video game characters move across the screen without changing size or shape? 🎮 That's the magic of congruence transformations! This tutorial is a comprehensive review of...

TermDefinitionExample Congruence Transformation (Isometry)A transformation that preserves distance, angle measure, and orientation. The pre-image (original figure) and the image (transformed figure) are congruent.Sliding a triangle 5 units to the right. The new triangle is identical in size and shape to the original. TranslationA transformation that 'slides' every point of a figure the same distance in the same direction.Triangle ABC with vertices A(1,2), B(3,5), C(4,1) is translated 3 units right and 2 units down. The new vertices are A'(4,0), B'(6,3), C'(7,-1). ReflectionA transformation that 'flips' a figure across a line, called the line of reflection, creating a mirror image.Reflecting the point P(2, 3) across the x-axis results in the image P'...

Translation Rule T_{a,b}(x, y) = (x + a, y + b) Use this to translate a point 'a' units horizontally and 'b' units vertically. If 'a' is positive, move right; negative, move left. If 'b' is positive, move up; negative, move down. Reflection Rules Across x-axis: R_{x-axis}(x, y) = (x, -y) \\ Across y-axis: R_{y-axis}(x, y) = (-x, y) \\ Across line y = x: R_{y=x}(x, y) = (y, x) Use these rules to find the coordinates of a point after it has been reflected over a specific line. The most common lines of reflection are the axes and the line y=x. Rotation Rules (about the origin) 90° counterclockwise: R_{90°}(x, y) = (-y, x) \\ 180°: R_{180°}(x, y) = (-x, -y) \\ 270° counterclockwise: R_{270°}(x, y) = (y, -x) Use these rules for coun...