Question

are the following triangles similar?
yes, the triangles were reflected and then dilated, so the shape is preserved.
yes, the triangles were translated and then dilated, so the shape is preserved.
no, the triangles are different sizes so they cannot be similar.
yes, the triangles were rotated and then dilated, so the shape is preserved.

Explanation:

Step1: Understand similarity of triangles

Similar triangles have the same shape (corresponding angles are equal) and their sides are in proportion. Transformations like reflection, translation, rotation are rigid - body transformations (they preserve shape and size). Dilation is a transformation that changes the size but preserves the shape (if the scale factor is non - zero).

Step2: Analyze each option

• Option 1 (Green): Reflection (a rigid - body transformation) followed by dilation (changes size but preserves shape). So the triangles are similar.
• Option 2 (Blue): Translation (a rigid - body transformation) followed by dilation (changes size but preserves shape). So the triangles are similar. But looking at the grid, we can observe the orientation change which is not just a translation.
• Option 3 (Orange): Different sizes (due to dilation) does not mean they are not similar. Similar triangles can have different sizes (as long as the ratio of corresponding sides is constant).
• Option 4 (Cyan): Rotation (a rigid - body transformation) followed by dilation (changes size but preserves shape). We can check the side - length ratios. Let's assume the smaller triangle has side lengths \(a,b,c\) and the larger one has side lengths \(ka,kb,kc\) (where \(k\) is the scale factor of dilation).

If we assume the coordinates of vertices (using the grid):
Let the vertices of the smaller triangle be \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) and of the larger triangle be \((x_1',y_1')\), \((x_2',y_2')\), \((x_3',y_3')\). Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we can find that the sides of the larger triangle are twice the sides of the smaller triangle (assuming a scale factor \(k = 2\)). And rotation (which can be verified by the change in orientation of the triangle with respect to the grid) is a part of the transformation.

Answer:

D. Yes, the triangles were rotated and then dilated, so the shape is preserved.