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.