Looking at the arrangement of triangles in the image, the problem is about identifying which triangles are translations of triangle 1 and which are not, based on their orientation.

Explanation:

1. Translations mean that the triangle must have the same shape and size, as well as the same orientation, as triangle 1. This can involve shifting the triangle horizontally or vertically, but it should not be rotated or reflected.

2. Non-Translations indicate that the triangle has been rotated or flipped, meaning its orientation is no longer the same as triangle 1.

Analysis of Orientations:

• Triangle 1 is positioned with its flat side (base) at the bottom and its vertex (point) at the top.
• The other triangles need to be evaluated based on whether they share this orientation.

Choices:

a. For translations (same orientation):

• Triangles in rows 1, 3, and 5 (numbered 1, 2, 3, 4, 5, 6, and 7) have the same orientation as triangle 1.
• These triangles are simply shifted horizontally or vertically.

b. For non-translations (rotated triangles):

• Triangles in rows 2 and 4 (numbered 9, 10, 11, 12, 13, 14, and 15) are flipped upside down. Their base is on top, so these triangles are not translations of triangle 1.

Filling in the blanks:

• a. The 1, 2, 3, 4, 5, 6, and 7 numbered triangles are translations of triangle 1.
• b. The 9, 10, 11, 12, 13, 14, and 15 numbered triangles are not translations of triangle 1 because the base side is on the "top" rather than on the "bottom." The 9, 10, 11, 12, 13, 14, and 15 numbered triangles are, however, reflections of one another.

Would you like more details, or do you have any questions about the reasoning?

Follow-up Questions:

1. How can we formally define translation and reflection in geometry?
2. What other transformations can be applied to shapes besides translation and reflection?
3. Can you identify specific triangles that are reflections of each other in this arrangement?
4. What happens if we rotate triangle 1 by 180 degrees? Would any triangles then match?
5. How would you describe the symmetry of the entire arrangement of triangles?

Tip:

In geometry, translations preserve the orientation, while reflections and rotations change it. Keep this in mind when identifying transformations.