The problem involves transforming triangle $\triangle ABC$ with given coordinates of $A(3, 1)$, $B(1, 8)$, and $C(7, 2)$ to a new triangle $\triangle A'B'C'$, as shown on the graph.

Step-by-step analysis:

• Original points:

  • $A(3, 1)$
  • $B(1, 8)$
  • $C(7, 2)$

• New points (after transformation):

  • From the graph, we can estimate the transformed points as:
    • $A'(-3, -1)$
    • $B'(-1, -8)$
    • $C'(-7, -2)$

By comparing the coordinates of $A$, $B$, and $C$ with their transformed counterparts $A'$, $B'$, and $C'$, it seems that the transformation involves reflection.

What type of reflection?

The transformation from $(x, y)$ to $(-x, -y)$ suggests a reflection across the origin, or equivalently, a rotation of 180° about the origin.

Answer:

The transformation that occurred is rotation by 180° about the origin.

Would you like more details or further clarifications?

Here are 5 related questions to deepen your understanding:

1. What is the general rule for rotating a point by 180° around the origin?
2. How would the transformation differ if it were a reflection over the x-axis instead?
3. Can you identify the coordinates of the new triangle after a reflection over the y-axis?
4. What are the effects of a 90° rotation about the origin on the triangle's coordinates?
5. How can you determine the center of rotation if it’s not the origin?

Tip: When analyzing transformations, always check for consistent changes in the coordinates to identify the transformation rule.