Triangle ABC, with vertices A(2, 4), B(4, 1), and C(6, 2), and triangle DEF, with vertices D(0, −4), E(−3, −1), and F(−4, −2), are plotted on a coordinate plane. Triangle ABC is rotated clockwise 180° about the origin and then translated 2 units to the right. Determine if triangles ABC and DEF are congruent. Choose the correct answers.

To determine if triangles ABC and DEF are congruent, we need to examine the transformation applied to triangle ABC and compare its new coordinates with those of triangle DEF.

### Step 1: Rotate triangle ABC by 180° clockwise about the origin.
A 180° clockwise rotation about the origin has the effect of flipping the signs of both the x- and y-coordinates of each point.

For each vertex of triangle ABC:
- \( A(2, 4) \rightarrow A'(-2, -4) \)
- \( B(4, 1) \rightarrow B'(-4, -1) \)
- \( C(6, 2) \rightarrow C'(-6, -2) \)

### Step 2: Translate the triangle 2 units to the right.
A translation of 2 units to the right means adding 2 to the x-coordinates of each vertex.

After the translation:
- \( A'(-2, -4) \rightarrow A''(0, -4) \)
- \( B'(-4, -1) \rightarrow B''(-2, -1) \)
- \( C'(-6, -2) \rightarrow C''(-4, -2) \)

Now, the new coordinates of triangle ABC are:
- \( A''(0, -4) \)
- \( B''(-2, -1) \)
- \( C''(-4, -2) \)

### Step 3: Compare the new coordinates of triangle ABC with triangle DEF.
The vertices of triangle DEF are:
- \( D(0, -4) \)
- \( E(-3, -1) \)
- \( F(-4, -2) \)

Notice that:
- \( A''(0, -4) = D(0, -4) \)
- \( C''(-4, -2) = F(-4, -2) \)

But:
- \( B''(-2, -1) \neq E(-3, -1) \)

### Conclusion:
Since not all corresponding vertices match exactly after the transformation, triangles ABC and DEF are not congruent. 

However, we should note that they might still be similar because their shapes could be the same but their sizes may differ.

Do you need more details or have any further questions?

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### Related Questions:
1. What is the difference between congruence and similarity in triangles?
2. How would the transformation change if the rotation was counterclockwise?
3. Can triangles be congruent if they are reflected instead of rotated?
4. How can we check for congruence using side lengths and angles?
5. What other transformations preserve congruence of triangles?

**Tip:** When checking for congruence, remember that both the shape and size must be exactly the same, even after any transformation.

Triangle ABC, with vertices A(2, 4), B(4, 1), and C(6, 2), and triangle DEF, with vertices D(0, −4), E(−3, −1), and F(−4, −2), are plotted on a coordinate plane. Triangle ABC is rotated clockwise 180° about the origin and then translated 2 units to the right. Determine if triangles ABC and DEF are congruent. Choose the correct answers.

Explore the congruence of triangle ABC after a 180° clockwise rotation and 2-unit translation compared to triangle DEF. This problem uses geometric transformations and the congruence criteria to determine if the two triangles are congruent. Suitable for Grades 8-10.

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