Question

What is the scale factor of the similar polygons below?
What is the scale factor of their perimeters?
What is the scale factor of their areas?

ABC ~ DEF

ABC = 4 6 8

DEF = 14 21 28

Answer 1

To determine the scale factor of the similar polygons \( ABC \) and \( DEF \), we need to compare the corresponding side lengths of the two polygons.

The sides of triangle \( ABC \) are \( 4 \), \( 6 \), and \( 8 \), while the sides of triangle \( DEF \) are \( 14 \), \( 21 \), and \( 28 \).

1. **Finding the scale factor for sides:**
- For side \( 4 \) (from \( ABC \)) and side \( 14 \) (from \( DEF \)):
\[
\text{Scale Factor} = \frac{14}{4} = \frac{7}{2} = 3.5
\]
- For side \( 6 \) and side \( 21 \):
\[
\text{Scale Factor} = \frac{21}{6} = \frac{7}{2} = 3.5
\]
- For side \( 8 \) and side \( 28 \):
\[
\text{Scale Factor} = \frac{28}{8} = \frac{7}{2} = 3.5
\]

Thus, the scale factor of the similar polygons is \( \frac{7}{2} \) or \( 3.5 \).

2. **Scale factor of their perimeters:**
The scale factor of the perimeters is the same as the scale factor of the sides. Therefore, it is also \( 3.5 \).

3. **Scale factor of their areas:**
The scale factor of the areas of similar polygons is the square of the scale factor of their sides. Thus, we calculate:
\[
\text{Scale Factor of Areas} = \left( \frac{7}{2} \right)^2 = \frac{49}{4}
\]

In summary:
- The scale factor of the polygons is \( 3.5 \).
- The scale factor of their perimeters is \( 3.5 \).
- The scale factor of their areas is \( \frac{49}{4} \) or \( 12.25 \).