← Grade 8 Geometry: Transformations & Similarity
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Grade 8 Geometry: Transformations & Similarity
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Grade 8 Geometry: Transformations & Similarity — Flashcards
| Front | Back |
|---|---|
| Euclid's "Elements of Geometry" | A foundational geometry text based on self-evident axioms and straightedge/compass constructions, deducing all geometric knowledge through logic; formed the basis of geometric instruction for over 2000 years. |
| Congruent (Euclidean definition) | Two figures are congruent if one can be copied onto the other using straightedge and compass constructions. |
| The Parallel Postulate controversy | The Euclidean axiom about parallel lines (that never meet) was questioned because one can never verify two lines never intersect; this dilemma lasted ~2000 years until resolved in the 19th century. |
| Spherical and hyperbolic geometry | Non-Euclidean geometries discovered in the 19th century where lines either all eventually intersect (spherical) or almost all never intersect (hyperbolic). |
| Felix Klein | 19th-century mathematician who formalized a new concept of geometry based on transformations, broad enough to encompass Euclidean and non-Euclidean geometries. |
| Transformational geometry | A dynamic approach to geometry (developed by Klein) where a set of transformations is specified, and geometry studies properties that remain unchanged under those transformations. |
| Coordinate (vector) geometry | An approach to geometry using coordinates and linear algebra, allowing precise calculation of measures (e.g., using the Pythagorean theorem as the definition of length). |
| Rigid motion | A transformation of the plane that takes lines to lines and preserves the lengths of line segments and the measures of angles. |
| Translation | A type of rigid motion, also called a "shift," that moves every point the same distance in the same direction. |
| Reflection | A type of rigid motion, also called a "flip," that preserves lengths and angle measures. |
| Rotation | A type of rigid motion, also called a "turn," that preserves lengths and angle measures. |
| Congruent (transformational definition) | Two figures are congruent if there is a sequence of rigid motions that takes one figure onto the other. |
| Dilation | A transformation that preserves lines and angles but changes the scale of line segment lengths by a constant factor (scale factor r); has one fixed point called the center. |
| Center of dilation | The single fixed point of a dilation; all other points move away from or toward this point. |
| Scale factor (r) | The positive number by which a dilation multiplies all lengths; if r = 1, the dilation is the identity (no movement). |
| Identity transformation | A dilation with scale factor r = 1, in which no point moves. |
| Similar figures | Two figures are similar (same shape) if there is a combination of rigid motions and dilations that takes one figure to the other. |
| Same shape vs. same size and shape | Figures with the same shape but different size are similar; figures with both the same shape and size (movable onto each other by rigid motion) are congruent. |
| Sum of two sides of a triangle | The sum of the lengths of any two sides of a triangle is greater than the length of the third side. |
| Perpendicular lines | Two lines that intersect such that all angles formed at the intersection point have the same measure. |
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