Question

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

a=(-2,-2)
b=(0,0)
c=(2,-4)

Answer 1

To find the new triangle that is dilated from the original triangle with the given vertices \( a = (-2, -2) \), \( b = (0, 0) \), and \( c = (2, -4) \) using a scale factor of \( 2.5 \) and a center of dilation at the origin, we can apply the dilation transformation to each vertex.

The formula for dilation with a center at the origin and a scale factor \( k \) is given by:

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

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

Using a scale factor \( k = 2.5 \):

1. **Dilation of point \( a = (-2, -2) \)**:
\[
a' = (2.5 \cdot -2, 2.5 \cdot -2) = (-5, -5)
\]

2. **Dilation of point \( b = (0, 0) \)**:
\[
b' = (2.5 \cdot 0, 2.5 \cdot 0) = (0, 0)
\]

3. **Dilation of point \( c = (2, -4) \)**:
\[
c' = (2.5 \cdot 2, 2.5 \cdot -4) = (5, -10)
\]

So the coordinates of the new triangle after dilation are:

- \( a' = (-5, -5) \)
- \( b' = (0, 0) \)
- \( c' = (5, -10) \)

Thus, the new triangle vertices are:
\[
\text{New triangle: } a' = (-5, -5), b' = (0, 0), c' = (5, -10)
\]