Triangles ABC, With vertices A(2,4), B(4,1), and C(6,2), and triangle DEF, With vertices D(0,-4),and E(-3,-1), and F(-4,-2), are plotted on a coordinate plane. Triangle ABC is rotated clockwise 180 degrees about the origin and than translated 2 Units to the right. Determine if triangles ABC and DEF are congruent.

To determine whether triangles ABC and DEF are congruent, we need to verify if their corresponding sides are equal in length after applying the given transformations to triangle ABC. Let's break this problem down step by step:

### Step 1: Find the vertices of triangle ABC after a 180-degree clockwise rotation about the origin.

A rotation of 180 degrees clockwise around the origin transforms any point \((x, y)\) into \((-x, -y)\). Applying this to each vertex of triangle ABC:

- Point A(2,4) becomes A'(-2,-4)
- Point B(4,1) becomes B'(-4,-1)
- Point C(6,2) becomes C'(-6,-2)

### Step 2: Translate the new triangle A'B'C' 2 units to the right.

A translation 2 units to the right increases the x-coordinate of each point by 2:

- Point A'(-2,-4) becomes A''(0,-4)
- Point B'(-4,-1) becomes B''(-2,-1)
- Point C'(-6,-2) becomes C''(-4,-2)

So, the vertices of the transformed triangle are \( A''(0,-4), B''(-2,-1), C''(-4,-2) \).

### Step 3: Compare the vertices of the transformed triangle with those of triangle DEF.

The vertices of triangle DEF are \( D(0,-4), E(-3,-1), F(-4,-2) \). 

Now, compare the vertices of triangle DEF with the transformed triangle A''B''C'':
- \( A''(0,-4) \) matches \( D(0,-4) \).
- \( B''(-2,-1) \) does not match \( E(-3,-1) \); there's a 1-unit difference in the x-coordinate.
- \( C''(-4,-2) \) matches \( F(-4,-2) \).

### Step 4: Check side lengths of both triangles.

The side lengths need to be checked to ensure the triangles are congruent. Let's calculate the side lengths of each triangle.

For triangle ABC:
- Length of AB: \( \sqrt{(4-2)^2 + (1-4)^2} = \sqrt{4 + 9} = \sqrt{13} \)
- Length of BC: \( \sqrt{(6-4)^2 + (2-1)^2} = \sqrt{4 + 1} = \sqrt{5} \)
- Length of CA: \( \sqrt{(6-2)^2 + (2-4)^2} = \sqrt{16 + 4} = \sqrt{20} \)

For triangle DEF:
- Length of DE: \( \sqrt{(0-(-3))^2 + (-4-(-1))^2} = \sqrt{9 + 9} = \sqrt{18} \)
- Length of EF: \( \sqrt{(-3-(-4))^2 + (-1-(-2))^2} = \sqrt{1 + 1} = \sqrt{2} \)
- Length of FD: \( \sqrt{(0-(-4))^2 + (-4-(-2))^2} = \sqrt{16 + 4} = \sqrt{20} \)

### Step 5: Conclusion

The side lengths of the transformed triangle A''B''C'' and triangle DEF are not the same, so the two triangles are **not congruent**. Despite two vertices matching, the corresponding sides differ in length, which rules out congruence.

Would you like more details or have any questions?

Here are five related questions you might find useful:
1. How do you calculate the distance between two points on a coordinate plane?
2. What are the properties of congruent triangles?
3. How do different types of transformations (like rotations and translations) affect the coordinates of a shape?
4. How do you determine if two triangles are similar instead of congruent?
5. How can we use transformations to map one figure onto another?

**Tip:** When working with transformations, always check both the angles and the side lengths to ensure congruence.

Triangles 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 degrees about the origin and then translated 2 units to the right. Determine if triangles ABC and DEF are congruent.

Explore the congruence of triangles ABC and DEF through transformations, including a 180-degree rotation and translation on the coordinate plane. This detailed solution covers geometric concepts and congruence criteria, ideal for students in Grades 8-10.

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