To find the final coordinates of the triangle after performing the dilation, we follow these steps:

Step 1: Dilation Formula

The dilation formula is given by: $(x', y') = (x_0 + k(x - x_0), y_0 + k(y - y_0))$ Where:

• $(x, y)$ are the original coordinates of the point.
• $(x', y')$ are the new coordinates of the point after dilation.
• $(x_0, y_0)$ is the center of dilation.
• $k$ is the scale factor.

Here:

• The center of dilation is $(1, 4)$.
• The scale factor $k = 2$.
• We need to apply this formula to each of the three vertices of the triangle: $(2, 7)$, $(2, 3)$, and $(5, 3)$.

Step 2: Apply the Dilation Formula

For point $(2, 7)$:

• $x_0 = 1, y_0 = 4, x = 2, y = 7$ $x' = 1 + 2(2 - 1) = 1 + 2 = 3$ $y' = 4 + 2(7 - 4) = 4 + 6 = 10$ The new coordinates are $(3, 10)$.

For point $(2, 3)$:

• $x_0 = 1, y_0 = 4, x = 2, y = 3$ $x' = 1 + 2(2 - 1) = 1 + 2 = 3$ $y' = 4 + 2(3 - 4) = 4 - 2 = 2$ The new coordinates are $(3, 2)$.

For point $(5, 3)$:

• $x_0 = 1, y_0 = 4, x = 5, y = 3$ $x' = 1 + 2(5 - 1) = 1 + 8 = 9$ $y' = 4 + 2(3 - 4) = 4 - 2 = 2$ The new coordinates are $(9, 2)$.

Final Coordinates

After the dilation, the new coordinates of the triangle are:

• $(3, 10)$
• $(3, 2)$
• $(9, 2)$

Let me know if you'd like more details or have any further questions!

Related Questions:

1. How would the dilation change if the scale factor were 0.5 instead of 2?
2. How do you find the center of dilation if you only know the original and dilated points?
3. What happens if the center of dilation is at the origin, $(0, 0)$?
4. Can dilation be performed on non-point objects like lines or circles? How?
5. How does the direction of the scale factor (positive vs. negative) affect the dilation?

Tip:

When dilating with a center other than the origin, it’s helpful to first find the vector from the center to each point, then multiply this vector by the scale factor. This approach simplifies visualizing the dilation.