Translations, Reflections and RotationsEdit
Before we continue, you need to know what Translations, Reflections, and Rotations are. Let us start with the following image of a triangle.
- Translation means moving the image horizontally (along the x axis) or vertically (along the y axis).
The image above was translated a few units horizontally (along the x axis) and a few units vertically (along the y axis).
- Reflection means flipping the image either over the x axis (a horizontal line) or over the y axis (a vertical line).
The image above was flipped over the x axis (a horizontal line).
- Rotation means moving the image from a pivot point.
The image above was rotated 90 degrees clockwise from a certain pivot point.
Notice that in all of the operations performed on the triangle above, none of the operations changed the angles of the triangle, or the lengths of any of the line segments. In all of the operations shown above, the only things that change are the location of the three points that make up the triangle.
Translations, reflections and rotations do not fundamentally change a shape. Terms for shapes that undergo any of these transformations are covered in the next section.
Congruence and SimilarityEdit
Intuitively, congruent shapes are shapes that are exactly the same. Technically speaking, two shapes are congruent if you can translate, rotate and/or reflect one of them in such a way that it coincides exactly with the other shape. Hence, a shape may be translated, reflected or rotated and remain congruent to its counterpart.
These two triangles above are congruent, even though they are rotations of each other.
Similar shapes are shapes that, when scaled, are exactly the same. A shape may be translated, reflected or rotated and remain similar to its counterpart. In a sense, similar shapes are scale models of each other, that is, they are proportional.
These triangles are similar because when one is scaled down, reflected, and rotated, it becomes congruent with the other.
- Congruent Shapes - Shapes that coincide exactly when translated, reflected, and/or rotated.
- Similar Shapes - Shapes that coincide exactly when translated, reflected, rotated and/or scaled.
1) On graph paper, draw a triangle with the points (0, 0), (0, 15) and (15, 0). Draw the triangle reflected over the Y axis (the vertical axis). Draw the triangle translated up 25 units.
2) On graph paper, draw a square with the points (0, 0), (0, 15), (15, 15) and (15, 0). When you scale this shape by any amount, what happens to the point at (0, 0)?
3)On graph paper, draw a square with the points (0, 0), (0, 6), (6, 6) and (6, 0). When you scale this shape by 2, what happens to the point at (0, 6)?
- Geometry Main Page
- Geometry/Chapter 1 Definitions and Reasoning (Introduction)
- Geometry/Chapter 2 Proofs
- Geometry/Chapter 3 Logical Arguments
- Geometry/Chapter 4 Congruence and Similarity
- Geometry/Chapter 5 Triangle: Congruence and Similiarity
- Geometry/Chapter 6 Triangle: Inequality Theorem
- Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
- Geometry/Chapter 8 Perimeters, Areas, Volumes
- Geometry/Chapter 9 Prisms, Pyramids, Spheres
- Geometry/Chapter 10 Polygons
- Geometry/Chapter 11
- Geometry/Chapter 12 Angles: Interior and Exterior
- Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
- Geometry/Chapter 14 Pythagorean Theorem: Proof
- Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
- Geometry/Chapter 16 Constructions
- Geometry/Chapter 17 Coordinate Geometry
- Geometry/Chapter 18 Trigonometry
- Geometry/Chapter 19 Trigonometry: Solving Triangles
- Geometry/Chapter 20 Special Right Triangles
- Geometry/Chapter 21 Chords, Secants, Tangents, Inscribed Angles, Circumscribed Angles
- Geometry/Chapter 22 Rigid Motion
- Geometry/Appendix A Formulae
- Geometry/Appendix B Answers to problems
- Appendix C. Geometry/Postulates & Definitions
- Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry