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Rigid motion refers to the transformation of a geometric object that preserves its shape and size, meaning distances and angles remain unchanged. It includes transformations like translations, rotations, and reflections, which are fundamental in geometry and physics for analyzing the movement and congruence of shapes.
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Translation is the process of converting text or speech from one language into another, ensuring that the meaning and context are preserved. It requires a deep understanding of both the source and target languages, as well as cultural nuances and idiomatic expressions.
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Rotation refers to the circular movement of an object around a center or an axis. This fundamental concept is pivotal in various fields, including physics, engineering, and mathematics, where it describes phenomena ranging from the Earth's rotation to the angular momentum of particles.
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Reflection is the process by which light or other waves bounce back from a surface, allowing us to see objects and perceive their colors. It is governed by the laws of physics, specifically the law of reflection, which states that the angle of incidence is equal to the angle of reflection.
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An isometry is a transformation in geometry that preserves distances between points, meaning the original shape and the transformed shape are congruent. Isometries include translations, rotations, reflections, and glide reflections, and are fundamental in understanding symmetry and rigid motions in Euclidean spaces.
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Congruence refers to the idea that two figures or objects are identical in shape and size, meaning they can be perfectly overlapped. In mathematics, congruence is a fundamental concept in geometry and number theory, where it describes figures that are equivalent under rigid transformations or numbers that have the same remainder when divided by a given modulus.
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Euclidean geometry is a mathematical system attributed to the ancient Greek mathematician Euclid, which describes the properties and relations of points, lines, surfaces, and solids in two and three dimensions. It is based on five postulates, including the famous parallel postulate, which forms the foundation for much of classical geometry taught in schools today.
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Symmetry refers to a balanced and proportionate similarity found in two halves of an object, which can be divided by a specific plane, line, or point. It is a fundamental concept in various fields, including mathematics, physics, and art, where it helps to understand patterns, structures, and the natural order.
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An affine transformation is a linear mapping method that preserves points, straight lines, and planes, allowing for operations like rotation, scaling, translation, and shearing. It is widely used in computer graphics, image processing, and geometric modeling to manipulate objects while maintaining their relative geometric properties.
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A transformation matrix is a mathematical tool used to perform linear transformations on vectors in a given space, such as scaling, rotating, or translating them. It is fundamental in computer graphics, physics simulations, and engineering applications for manipulating spatial data efficiently and consistently.
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A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars, adhering to specific axioms such as associativity, commutativity, and distributivity. It provides the foundational framework for linear algebra, enabling the study of linear transformations, eigenvalues, and eigenvectors, which are crucial in various fields including physics, computer science, and engineering.
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Congruent transformation refers to a geometric operation that alters the position or orientation of a shape without changing its size or shape. It preserves distances and angles, ensuring that the original and transformed figures are congruent, meaning they are identical in form and dimension.
Mathematical translation movement involves shifting a geometric figure in a plane without altering its shape, size, or orientation. This transformation is described by a vector that indicates the direction and distance of the movement, maintaining the congruence of the original figure and its image.
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Triangle congruence is a fundamental concept in geometry that states two triangles are congruent if their corresponding sides and angles are equal. This can be proven using specific criteria such as Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL) for right triangles.
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A Euclidean transformation is a geometric transformation that preserves distances and angles, ensuring that the shape and size of geometric figures remain unchanged. It includes operations such as translations, rotations, and reflections, which are fundamental in maintaining the congruence of figures in Euclidean space.
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Isometries are transformations that preserve distances between points in a geometric space, ensuring that the shape and size of figures remain unchanged. They are fundamental in understanding symmetry and congruence in geometry, often used in fields like physics and computer graphics for maintaining structural integrity during transformations.
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The Side-Side-Side (SSS) criterion is a fundamental principle in geometry used to prove the congruence of two triangles, stating that if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. This criterion relies solely on the lengths of the sides, making it a powerful tool for proving congruence without needing to consider angles.
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