# A triangle is dilated using a scale factor of 4. The image is then dilated using a scale factor of 3. What scale factor could you use to dilate the original triangle to get the final image?

Nov 27, 2015

For perimeter, the scale factor would be $\frac{a}{b}$ for enlargement, where $a$ is the perimeter of the dilated triangle and $b$ is the perimeter of the smaller triangle.
For area, the scale factor would be $\sqrt{\frac{x}{y}}$ for enlargement, where $x$ is the area of the dilated triangle and $y$ is the area of the smaller triangle.

#### Explanation:

$\rightarrow$If you are referring to perimeter, the scale factor would be 12.

Example:
Perimeter = $15$ $c m$
Dilated by scale factor $4 = 60$ $c m$
Dilated by scale factor $3 = 180$ $c m$

• To get from $15$ $c m$ to $180$ $c m$, you would use a scale factor of $180 \div 15 = 12$.
• The scale factor is equal to the change in perimeter between the smaller and dilated triangle.
For example, $5$ $c m$$\cdot$$12$ $c m = 180$ $c m$

$\rightarrow$If you are referring to area, the scale factor would also be 12.

Example:
Area = $4 \sqrt{3}$ $c {m}^{2}$
Dilated by scale factor $4 \approx 110.85$ $c {m}^{2}$
Dilated by scale factor $3 \approx 997.66$ $c {m}^{2}$

• To get from $4 \sqrt{3}$ $c {m}^{2}$ to $997.66$ $c {m}^{2}$, you would use a scale factor of $\sqrt{997.66 \div 4 \sqrt{3}} \approx 12$
• The only difference between the perimeter and area is that the change in area between two similar triangles is not equal to the scale factor.
For example, $4 \sqrt{3}$ $c m$$\cdot$$12$ $c m \ne 997.56$
• To get the scale factor for the area between two triangles, you must square root the change in area between the triangles.