Similarity

Similarity in Geometry

Basic Concept:

  • Similarity in geometry refers to the correlation between two shapes that are identical in proportion and angles, yet vary in size.
  • Different sizes of the same shape are known as similar shapes. They maintain the same shape as they grow or shrink.
  • When one shape can become another after a rescale, resize, dilate or similarity transformation, the shapes are said to be similar.

Properties of Similar Shapes:

  • In two similar figures, corresponding angles are equal and corresponding sides are proportional.
  • The scale factor is the ratio of the lengths of corresponding sides of two similar shapes. For example, if the scale factor is 1:2, every length on the smaller shape is half the length of the corresponding length on the bigger shape.
  • Similarity is transitive; if shape A is similar to shape B, and shape B is similar to shape C, then shape A is similar to shape C.

Similar Triangles:

  • Triangles are similar if their corresponding sides are in proportion and their corresponding angles are equal.
  • The AAA (Angle-Angle-Angle) similarity criterion states that if two triangles have their corresponding angles equal, then they are similar.
  • The SAS (Side-Angle-Side) similarity criterion states if two triangles have one of their angles equal and the sides containing these angles in proportion, then the triangles are similar.
  • The SSS (Side-Side-Side) similarity criterion states if the three sides of a triangle are in proportion to the three sides of another triangle, then these two triangles are similar.

Applications of Similarity:

  • The principle of similarity is used in a variety of real-world contexts such as architecture, art, photography and satellite imagery.
  • Similarity rules are applied to calculate distances and heights in maps and diagrams without actual measurement, using proportions and ratios.
  • Theorems related to similarity, such as the Pythagorean theorem and trigonometric ratios, are fundamental to many branches of mathematics, physics and engineering.

Carefully understanding the concept of geometric similarity and its practical applications can enhance your problem-solving capabilities in various mathematical and real-world contexts.