Similarity

In geometry, similarity refers to a relationship between two shapes where they have the same shape but may differ in size. Similar figures maintain proportional dimensions and identical angles.

Key Properties of Similar Figures

1. Same Shape:

   • Corresponding angles are equal.

   • Corresponding sides are proportional.

2. Proportionality:

   • The ratio of any two corresponding sides in similar figures is the same:
     Side 1 of Figure ASide 1 of Figure B=Side 2 of Figure ASide 2 of Figure BSide 1 of Figure BSide 1 of Figure A​=Side 2 of Figure BSide 2 of Figure A​

3. Scale Factor:

   • The ratio of the lengths of corresponding sides.

   • Used to calculate dimensions in one figure based on another.

4. Preserved Angles:

   • All corresponding angles in similar figures are congruent.

Tests for Similarity

1. AA (Angle-Angle) Criterion:

   • If two angles in one triangle are equal to two angles in another triangle, the triangles are similar.

2. SSS (Side-Side-Side) Criterion:

   • If the ratios of all three pairs of corresponding sides are equal, the triangles are similar.

3. SAS (Side-Angle-Side) Criterion:

   • If the ratios of two pairs of corresponding sides are equal, and the included angles are equal, the triangles are similar.

In similar triangles:

3. • All corresponding angles are equal.

   • Corresponding sides are proportional.

   • The ratio of perimeters equals the scale factor.

Key Formulas for Similarity

• Ratios of Sides:

  • If two figures are similar with a scale factor of 'k', then the ratio of their corresponding side lengths is also 'k'.

  • If side 'a' of the first figure corresponds to side 'a'' of the second figure, then:

    • a/a' = k

• Areas of Similar Figures:

  • If the scale factor of two similar figures is 'k', then the ratio of their areas is k².

  • If the area of the first figure is 'A' and the area of the second figure is 'A'', then:

    • A/A' = k²

• Volumes of Similar Figures:

  • If the scale factor of two similar figures is 'k', then the ratio of their volumes is k³.

  • If the volume of the first figure is 'V' and the volume of the second figure is 'V'', then:

    • V/V' = k³

In simpler terms:

• Sides: Scale factor is a direct relationship.

• Areas: Scale factor is squared.

• Volumes: Scale factor is cubed.

Example:

If two similar cubes have a scale factor of 2 (meaning one cube's sides are twice as long as the other's):

• Their side lengths are in a ratio of 2:1

• Their surface areas are in a ratio of 2² = 4:1

• Their volumes are in a ratio of 2³ = 8:1

Applications of Similarity

1. Maps and Scale Drawings:

   • Used in navigation, architecture, and planning.

2. Optics and Shadows:

   • Determining heights of objects using proportions (e.g., shadow lengths).

3. Modeling:

   • Creating scaled models of larger objects (e.g., airplanes, buildings).

4. Astronomy:

   • Calculating distances of stars using similar triangles.