Ratios, Proportions, and Similarity in Geometry

Ratio: A ratio serves to compare quantities.
Proportion: A proportion is defined as a statement that two ratios are equal.
Numerical Example of Proportion: An example of a proportion is the equation $\frac{3}{5} = \frac{6}{10}$ .
Direct Proportion: In a relationship of direct proportion, if one factor increases, the other factor also increases. Conversely, if one factor decreases, the other factor decreases as well.

General Definition: Similar triangles are defined as having the same shape, though they do not necessarily possess the same size.
Corresponding Angles: In similar triangles, the corresponding angles are equal to one another.
Corresponding Sides: The corresponding sides of similar triangles are proportional.
Vertex Ordering: The specific order of the letters is critical when matching corresponding vertices between figures.

Definition of Scale Factor: The scale factor is a value that indicates the extent to which a triangle has been enlarged or reduced.
Measuring Scale Factor: To calculate the scale factor, take the new length of a side and divide it by the original length of the corresponding side.
Scale Factor Formula: $\text{Scale Factor} = \frac{\text{New Length}}{\text{Original Length}}$
Numerical Example of Scale Factor:
- If a new length is given as 15 and the original linear length is given as 5, the calculation is $\frac{15}{5} = 3$ .
- In this scenario, the scale factor is 3.
Applying the Scale Factor: To find a new length, you must multiply the corresponding original length by the calculated scale factor.

Area Factor: There is a specific relationship between the linear scale factor and the area.
If the linear scale factor is represented by the variable $k$ , then the corresponding area factor is represented by $k^2$ ( $k$ to the power of two).