Question

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?

Answer 1

For perimeter, the scale factor would be `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(x/y)` for enlargement, where `x` is the area of the dilated triangle and `y` is the area of the smaller triangle.

Explanation:

`rarr`If you are referring to perimeter, the scale factor would be 12.

Example:
Perimeter = `15` `cm`
Dilated by scale factor `4 = 60` `cm`
Dilated by scale factor `3 = 180` `cm`

• To get from `15` `cm` to `180` `cm`, you would use a scale factor of `180-:15=12`.
• The scale factor is equal to the change in perimeter between the smaller and dilated triangle.
  For example, `5` `cm` `*` `12` `cm=180` `cm`

`rarr`If you are referring to area, the scale factor would also be 12.

Example:
Area = `4sqrt(3)` `cm^2`
Dilated by scale factor `4 ~~ 110.85` `cm^2`
Dilated by scale factor `3 ~~ 997.66` `cm^2`

• To get from `4sqrt(3)` `cm^2` to `997.66` `cm^2`, you would use a scale factor of `sqrt(997.66-:4sqrt(3))~~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, `4sqrt(3)` `cm` `*` `12` `cm!=997.56`
• To get the scale factor for the area between two triangles, you must square root the change in area between the triangles.