You find the scale factor ratio by dividing a dimension of the new shape (image) by the corresponding dimension of the original shape (pre-image). Essentially, it's the ratio of the 'after' size to the 'before' size.
Here's a breakdown:
Understanding Scale Factor
The scale factor indicates how much a shape has been enlarged or reduced.
- Scale factor > 1: Enlargement (the new shape is bigger).
- Scale factor < 1: Reduction (the new shape is smaller).
- Scale factor = 1: The shapes are congruent (same size).
The Formula
The basic formula is:
Scale Factor = (Dimension of New Shape) / (Dimension of Original Shape)
or
Scale Factor = (Image Dimension) / (Pre-image Dimension)
Steps to Calculate the Scale Factor
- Identify Corresponding Sides: Make sure you are comparing corresponding sides (sides in the same relative position) of the two shapes.
- Measure the Dimensions: Accurately measure the length of the corresponding sides in both shapes. The units must be the same (e.g., both in cm, both in inches).
- Apply the Formula: Divide the dimension of the new shape by the dimension of the original shape.
- Simplify (if possible): Express the scale factor as a simplified fraction or a decimal.
Examples
Example 1: Enlargement
- Original triangle side: 4 cm
- New triangle side: 8 cm
Scale Factor = 8 cm / 4 cm = 2
The scale factor is 2, meaning the new triangle is twice as big as the original.
Example 2: Reduction
- Original rectangle width: 10 inches
- New rectangle width: 5 inches
Scale Factor = 5 inches / 10 inches = 1/2 = 0.5
The scale factor is 1/2 or 0.5, meaning the new rectangle is half the size of the original.
Example 3: Same Size
- Original square side: 3 meters
- New square side: 3 meters
Scale Factor = 3 meters / 3 meters = 1
The scale factor is 1, meaning the new square is the same size as the original.
Importance of Corresponding Sides
It's crucial to use corresponding sides. Comparing non-corresponding sides will yield an incorrect scale factor. Corresponding sides are sides that occupy the same relative position in similar figures.
Conclusion
Finding the scale factor is a simple division problem, but it relies on accurately identifying and measuring corresponding sides. This ratio provides a clear indication of how much a shape has been enlarged or reduced in size.
