Question

If you dilate triangle ABC to form X'Y'Z' using a magnitude of 2 and a center of dilation at (2,0), what are the coordinates of the dilated triangle?



(1 point)
Responses

A' (2,12), B' (6,0), C' (12,8)
A' (2,12), B' (6,0), C' (12,8)

A' (.5,3), B' (1.5,0), C' (3,2)
A' (.5,3), B' (1.5,0), C' (3,2)

A' (0,12), B' (4,0), C' (10,8)
A' (0,12), B' (4,0), C' (10,8)

A' (1.5,3), B' (2.5,0), C' (4,2)
A' (1.5,3), B' (2.5,0), C' (4,2)
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Answer 1

a = (1,6),c=(6,4).b=(3,0)

Answer 2

To perform a dilation of triangle ABC with a center of dilation at (2,0) and a magnitude of 2, we can follow this formula for each vertex (x, y) of the triangle:

\[
(x', y') = (cx + k(x - c), cy + k(y - c))
\]

where (c, d) is the center of dilation, k is the scale factor (magnitude), and (x', y') are the coordinates of the dilated point.

Given:
- Center of dilation: \( (2, 0) \)
- Magnitude (scale factor, k): \( 2 \)

We can simplify the dilation formula for our specific case as follows:

1. Calculate the change in coordinates due to the dilation:
\[
x' = c + k(x - c) = 2 + 2(x - 2)
\]
\[
y' = 0 + 2(y - 0) = 2y
\]

Now, let's break down how this applies to the points of triangle ABC. We need the original coordinates of points A, B, and C. Since we don't have those specific coordinates provided, let's analyze the provided options instead, assuming we need to check which set results from a proper dilation.

Let’s check the coordinates given in the responses:

1. **Response A**: A' (2,12), B' (6,0), C' (12,8)
- For A' (2, 12):
\[
x' = 2 + 2(2 - 2) = 2 \quad \text{and} \quad y' = 0 + 2(12 - 0) = 24 \quad (\text{not consistent})
\]
- This response does not maintain the dilation requirement.

2. **Response B**: A' (0.5, 3), B' (1.5, 0), C' (3, 2)
- Check A' (0.5, 3):
\[
x' = 2 + 2(0.5 - 2) = -1 \quad \text{and} \quad y' = 2(3) = 6 \quad (\text{not consistent})
\]
- This response does not maintain the dilation requirement.

3. **Response C**: A' (0, 12), B' (4, 0), C' (10, 8)
- Check A' (0, 12):
\[
x' = 2 + 2(0 - 2) = -2 \quad \text{and} \quad y' = 2(12) = 24 \quad (\text{not consistent})
\]
- This response does not maintain the dilation requirement.

4. **Response D**: A' (1.5, 3), B' (2.5, 0), C' (4, 2)
- Check A' (1.5, 3):
\[
x' = 2 + 2(1.5 - 2) = 1 \quad \text{and} \quad y' = 2(3) = 6 \quad (\text{not consistent})
\]
- This response does not maintain the dilation requirement.

After checking all provided options, none of the proposed locations for the dilated triangle match the expected outcome based on the dilation rules. If you have the original coordinates of triangle ABC, you could provide them for accuracy, or we can assume a systematic exploration of the above responses is sufficient for the current context.

If triangle ABC's original coordinates are known, please share for a precise calculation.

Answer 3

Given the original coordinates of triangle ABC as:

- A = (1, 6)
- B = (3, 0)
- C = (6, 4)

we will apply the dilation with a center at (2, 0) and a magnitude of 2. Using the dilation formula for each vertex \((x, y)\):

\[
(x', y') = \left( c + k(x - c), d + k(y - d) \right)
\]

Here,
- Center of dilation (c, d) = (2, 0)
- Scale factor \( k = 2 \)

### Step-by-Step Calculation

1. **For point A (1, 6):**
\[
x' = 2 + 2(1 - 2) = 2 + 2(-1) = 2 - 2 = 0
\]
\[
y' = 0 + 2(6 - 0) = 0 + 2 \times 6 = 12
\]
\[
A' = (0, 12)
\]

2. **For point B (3, 0):**
\[
x' = 2 + 2(3 - 2) = 2 + 2(1) = 2 + 2 = 4
\]
\[
y' = 0 + 2(0 - 0) = 0 + 0 = 0
\]
\[
B' = (4, 0)
\]

3. **For point C (6, 4):**
\[
x' = 2 + 2(6 - 2) = 2 + 2(4) = 2 + 8 = 10
\]
\[
y' = 0 + 2(4 - 0) = 0 + 2 \times 4 = 8
\]
\[
C' = (10, 8)
\]

### Final Coordinates of the Dilated Triangle

Thus, the coordinates of the dilated triangle \( A'B'C' \) are:

- \( A' = (0, 12) \)
- \( B' = (4, 0) \)
- \( C' = (10, 8) \)

### Conclusion

Therefore, the correct response among the options is:
**A' (0, 12), B' (4, 0), C' (10, 8).**