To dilate triangle ABC to form triangle XYZ using a magnitude of 2 and a center of dilation at the point (2, 0), you'll follow these steps for each vertex of triangle ABC.
Dilation Formula
The formula for dilating a point \( P(x, y) \) from a center of dilation \( C(cx, cy) \) with a scale factor \( k \) is given by:
\[ P' = C + k(P - C) \]
Where:
- \( P' \) is the new point after dilation.
- \( C \) is the center of dilation.
- \( k \) is the scale factor.
- \( P \) is the original point.
Steps
- Identify the vertices of triangle ABC: Let's denote the vertices of triangle ABC as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).
- Apply the dilation formula to each vertex.
Example Calculation
Let's assume the vertices of triangle ABC are:
- \( A(1, 1) \)
- \( B(3, 1) \)
- \( C(2, 2) \)
The center of dilation is \( C(2, 0) \) and the scale factor \( k = 2 \).
For point A(1, 1): \[ P' = (2, 0) + 2((1, 1) - (2, 0)) \] \[ = (2, 0) + 2(-1, 1) \] \[ = (2, 0) + (-2, 2) = (0, 2) \]
For point B(3, 1): \[ P' = (2, 0) + 2((3, 1) - (2, 0)) \] \[ = (2, 0) + 2(1, 1) \] \[ = (2, 0) + (2, 2) = (4, 2) \]
For point C(2, 2): \[ P' = (2, 0) + 2((2, 2) - (2, 0)) \] \[ = (2, 0) + 2(0, 2) \] \[ = (2, 0) + (0, 4) = (2, 4) \]
Coordinates of Triangle XYZ
After applying the dilation to all vertices, the coordinates of triangle XYZ will be:
- \( X(0, 2) \)
- \( Y(4, 2) \)
- \( Z(2, 4) \)
You can apply this method to any points of triangle ABC to get the corresponding dilated triangle.