Similar Figures and Scale Factor

Similar Figures

Two figures are similar ($\sim$) if:

1. Corresponding angles are congruent
2. Corresponding sides are proportional

$\triangle ABC \sim \triangle DEF$

Scale Factor

The ratio of corresponding side lengths:

$k = \frac{\text{new length}}{\text{original length}}$

• $k > 1$: Enlargement
• $k < 1$: Reduction
• $k = 1$: Congruent

Triangle Similarity Criteria

AA (Angle-Angle)

If two angles of one triangle are congruent to two angles of another, the triangles are similar.

SSS Similarity

If all three pairs of corresponding sides are proportional: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$

SAS Similarity

If two pairs of corresponding sides are proportional AND the included angles are congruent.

Proportional Relationships

If $\triangle ABC \sim \triangle DEF$ with scale factor $k$:

| Measurement | Scale | |------------|-------| | Lengths | $\times k$ | | Perimeters | $\times k$ | | Areas | $\times k^2$ | | Volumes | $\times k^3$ |

Dilations

A transformation that produces similar figures: $(x, y) \to (kx, ky)$

Center at origin, scale factor $k$.

Parallel Lines and Similar Triangles

If a line parallel to one side of a triangle intersects the other two sides, it creates a smaller similar triangle.

Triangle Proportionality Theorem: $\frac{AD}{DB} = \frac{AE}{EC}$

Remember: Similar figures have the SAME shape but can be different SIZES. Congruent figures are similar with scale factor $k = 1$.