To dilate a triangle from a center of dilation, you can use the formula:
\[ X' = C_x + k(X - C_x) \] \[ Y' = C_y + k(Y - C_y) \]
where \((X, Y)\) are the coordinates of the original point, \((C_x, C_y)\) are the coordinates of the center of dilation, \(k\) is the magnitude of the dilation, and \((X', Y')\) are the coordinates of the dilated point.
In this case, the center of dilation is \((2, 0)\) and the magnitude of dilation is \(2\).
Now let's apply this to each of the vertices A, B, and C.
Coordinates of Vertex A (1, 6)
\[ X_A' = 2 + 2(1 - 2) = 2 + 2(-1) = 2 - 2 = 0 \] \[ Y_A' = 0 + 2(6 - 0) = 0 + 2(6) = 0 + 12 = 12 \] So, \(A'\) is at \((0, 12)\).
Coordinates of Vertex B (3, 0)
\[ X_B' = 2 + 2(3 - 2) = 2 + 2(1) = 2 + 2 = 4 \] \[ Y_B' = 0 + 2(0 - 0) = 0 + 2(0) = 0 + 0 = 0 \] So, \(B'\) is at \((4, 0)\).
Coordinates of Vertex C (6, 4)
\[ X_C' = 2 + 2(6 - 2) = 2 + 2(4) = 2 + 8 = 10 \] \[ Y_C' = 0 + 2(4 - 0) = 0 + 2(4) = 0 + 8 = 8 \] So, \(C'\) is at \((10, 8)\).
Conclusion
The coordinates of the dilated triangle \(X'Y'Z'\) are:
- \(A' = (0, 12)\)
- \(B' = (4, 0)\)
- \(C' = (10, 8)\)
Thus, the coordinates of the dilated triangle are: \[ (0, 12), (4, 0), (10, 8) \]
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