Question

Find the new triangle that undergoes a scale factor of 2.5 and a center of dilation at the origin

Answer 1

To find the new triangle after applying a dilation with a scale factor of 2.5 and a center of dilation at the origin, you will need the coordinates of the original triangle's vertices. Let’s assume the vertices of the original triangle are given as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).

The formula for dilating a point \( (x, y) \) by a scale factor \( k \) from the origin is:

\[
(x', y') = (k \cdot x, k \cdot y)
\]

where \( (x', y') \) are the coordinates of the new point after dilation.

### Steps to Calculate the New Coordinates:
1. **Identify the original vertices:** Let’s say our original triangle has the vertices \( A(1, 2) \), \( B(3, 4) \), and \( C(5, 1) \) as an example.
2. **Apply the scale factor of 2.5 to each vertex.**

For vertex \( A(x_1, y_1) = (1, 2) \):
\[
A' = (2.5 \cdot 1, 2.5 \cdot 2) = (2.5, 5)
\]

For vertex \( B(x_2, y_2) = (3, 4) \):
\[
B' = (2.5 \cdot 3, 2.5 \cdot 4) = (7.5, 10)
\]

For vertex \( C(x_3, y_3) = (5, 1) \):
\[
C' = (2.5 \cdot 5, 2.5 \cdot 1) = (12.5, 2.5)
\]

### Conclusion:
Thus, the coordinates of the new triangle after a dilation by a scale factor of 2.5 centered at the origin would be:

- \( A'(2.5, 5) \)
- \( B'(7.5, 10) \)
- \( C'(12.5, 2.5) \)

If you have specific coordinates for the original triangle's vertices, you would apply the same process to find the new triangle's vertices.