Let's break down the image to find the correct answer.

Analysis of the Triangles $T$ and $T'$

1. Triangle $T$ is larger and positioned to the left of the origin.
2. Triangle $T'$ is smaller and positioned to the right of the origin.

We need to check if $T$ can be transformed into $T'$ by a combination of dilation and translation.

Step 1: Check the Scale Factor

• Dilation is a transformation that changes the size of a figure by a scale factor.
• Comparing $T$ and $T'$, we can see that $T'$ is smaller than $T$.
• It looks like $T'$ is approximately half the size of $T$, suggesting a scale factor of $\frac{1}{2}$.

Step 2: Check the Translation

After applying a dilation with a scale factor of $\frac{1}{2}$, the resulting figure can be translated:

• $T'$ is located about 3 units to the right of the origin compared to the scaled version of $T$.

Step 3: Conclusion

Given the observations, the true statement should reflect:

1. Dilation by a scale factor of $\frac{1}{2}$.
2. Translation 3 units to the right.
3. The triangles remain similar because dilation preserves similarity.

Correct Answer

The fourth option is correct:

• ✅ "Since $T$ can be mapped to $T'$ using a dilation with a scale factor of $\frac{1}{2}$ about the origin, followed by a translation 3 units to the right, the triangles are similar."

Would you like a detailed explanation of transformations or have any questions?

Related Questions:

1. What is the definition of dilation and how does it affect shapes?
2. How do you determine the scale factor of a dilation?
3. What does it mean for two triangles to be similar?
4. How do translations affect the coordinates of a figure?
5. Can a dilation ever result in congruent figures?

Tip:

Dilation scales a figure by multiplying all its coordinates by the scale factor, while translations shift a figure without changing its size or shape.