Final answer:
In summary, the area of the dilated image of a rectangle is found by squaring the scale factor k and multiplying it by the original area. For a scale factor of 2, the new area would be 48 square units.
Step-by-step explanation:
When a rectangle with an area of 12 square units is dilated by a scale factor of k, the area of the resulting image can be determined using the relationship that the area of the dilated figure is equal to the square of the scale factor multiplied by the original area.
For example, if the scale factor is 2, as in the provided information, you would calculate the new area as follows:
- Original area = 12 square units
- Scale factor (k) = 2
- New area = k2 × original area = 22 × 12 = 4 × 12 = 48 square units
The rule of area and scale factor states that for similar figures, the ratio of their areas is the square of the scale factor. This is why the area of the dilated image is k2 times the original area. Applying this concept, you can find the area of the image for any given value of k.