under which transformation is the area of a triangle not equal to the area of its image?

i think the answer is dilation

You are correct! Under a dilation, the area of a triangle is not equal to the area of its image.

To understand this, let's first review what a dilation is. A dilation is a transformation that resizes an object without changing its shape. It is performed by multiplying each coordinate of the object by a scale factor, which can be greater than 1 (for an enlarged image), equal to 1 (for an unchanged image), or between 0 and 1 (for a reduced image).

When a dilation is applied to a triangle, all three sides of the triangle are scaled by the same factor. This means that the lengths of the sides are changed, but the shape of the triangle is preserved. However, the area of a triangle is determined by the length of its base and its height.

Under a dilation, the lengths of the sides are changed, but the base and height of the triangle are also scaled by the same factor. Since the area of a triangle is proportional to its base times its height, the scale factor affects both the base and the height, resulting in a different area for the image of the triangle.

Therefore, under a dilation, the area of a triangle is not equal to the area of its image.