The coordinates of the preimage (Triangle) are as follows:  A(2,4), B(4, 3), C(2, -1)
Enlarge the preimage about point D(3,-1) by scale factor 2 to obtain the image.Show the working for all vertices of the image.

To enlarge a triangle about a point \( D(3, -1) \) by a scale factor of 2, we will apply the following steps to each vertex of the triangle:

### General formula for dilation:
\[
(x', y') = (D_x + k(x - D_x), D_y + k(y - D_y))
\]
Where:
- \((x', y')\) are the coordinates of the image point,
- \(D_x\) and \(D_y\) are the coordinates of the dilation center \( D(3, -1) \),
- \(k\) is the scale factor (which is 2 in this case),
- \((x, y)\) are the coordinates of the preimage point.

### Step-by-step dilation for each vertex:

#### 1. Vertex \( A(2, 4) \):
- Preimage coordinates: \( A(2, 4) \)
- Dilation center: \( D(3, -1) \)
- Scale factor: \( k = 2 \)

Apply the dilation formula:
\[
x'_A = 3 + 2(2 - 3) = 3 + 2(-1) = 3 - 2 = 1
\]
\[
y'_A = -1 + 2(4 - (-1)) = -1 + 2(4 + 1) = -1 + 2(5) = -1 + 10 = 9
\]
Thus, the new coordinates of \( A' \) are \( A'(1, 9) \).

#### 2. Vertex \( B(4, 3) \):
- Preimage coordinates: \( B(4, 3) \)
- Dilation center: \( D(3, -1) \)
- Scale factor: \( k = 2 \)

Apply the dilation formula:
\[
x'_B = 3 + 2(4 - 3) = 3 + 2(1) = 3 + 2 = 5
\]
\[
y'_B = -1 + 2(3 - (-1)) = -1 + 2(3 + 1) = -1 + 2(4) = -1 + 8 = 7
\]
Thus, the new coordinates of \( B' \) are \( B'(5, 7) \).

#### 3. Vertex \( C(2, -1) \):
- Preimage coordinates: \( C(2, -1) \)
- Dilation center: \( D(3, -1) \)
- Scale factor: \( k = 2 \)

Apply the dilation formula:
\[
x'_C = 3 + 2(2 - 3) = 3 + 2(-1) = 3 - 2 = 1
\]
\[
y'_C = -1 + 2(-1 - (-1)) = -1 + 2(0) = -1 + 0 = -1
\]
Thus, the new coordinates of \( C' \) are \( C'(1, -1) \).

### Final Coordinates of the Image:
- \( A'(1, 9) \)
- \( B'(5, 7) \)
- \( C'(1, -1) \)

This gives the new triangle after dilation with a scale factor of 2 about the point \( D(3, -1) \).

Would you like any further details or clarifications?

Here are five related questions to deepen your understanding:
1. What would the image coordinates be if the scale factor was 3 instead of 2?
2. How does changing the center of dilation affect the image?
3. Can you explain why the point \( C \) remains on the same horizontal line as the center of dilation?
4. How would a negative scale factor affect the triangle’s image?
5. What is the relationship between the areas of the preimage and the image after dilation?

**Tip:** When performing dilation, always remember that points closer to the center of dilation move less, while points farther away move more dramatically.

The coordinates of the preimage (Triangle) are as follows: A(2,4), B(4, 3), C(2, -1). Enlarge the preimage about point D(3,-1) by scale factor 2 to obtain the image. Show the working for all vertices of the image.

Learn how to perform dilation of a triangle with vertices A(2,4), B(4,3), and C(2,-1) about the point D(3,-1) by a scale factor of 2. This step-by-step guide covers key concepts in geometry, including dilation formulas and transformations. Suitable for students in Grades 9-12.

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