A Polygon has 4 vertices located as A (20, 10), B(60, 10), 6(60, 30) D(20,30) calculate the vest- -ces after applying a Tranformation matrix double the size ot polygon with As located on the same place
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Answer:
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Answer:To create a transformation matrix to double the size of the polygon with point A located at the same place, you can follow these steps:
1. Calculate the centroid of the polygon:
- Find the average of the x-coordinates of the vertices: (20 + 60 + 60 + 20) / 4 = 40
- Find the average of the y-coordinates of the vertices: (10 + 10 + 30 + 30) / 4 = 20
The centroid of the polygon is located at (40, 20).
2. Translate the polygon so that the centroid is at the origin (0, 0):
- Subtract the centroid coordinates from each vertex:
A' = (20 - 40, 10 - 20) = (-20, -10)
B' = (60 - 40, 10 - 20) = (20, -10)
C' = (60 - 40, 30 - 20) = (20, 10)
D' = (20 - 40, 30 - 20) = (-20, 10)
3. Scale the polygon by a factor of 2:
- Multiply the x and y coordinates of each vertex by 2:
A'' = 2 * (-20, -10) = (-40, -20)
B'' = 2 * (20, -10) = (40, -20)
C'' = 2 * (20, 10) = (40, 20)
D'' = 2 * (-20, 10) = (-40, 20)
4. Translate the polygon back to the original position by adding the centroid coordinates:
- Add the centroid coordinates to each vertex:
A''' = (-40 + 40, -20 + 20) = (0, 0)
B''' = (40 + 40, -20 + 20) = (80, 0)
C''' = (40 + 40, 20 + 20) = (80, 40)
D''' = (-40 + 40, 20 + 20) = (0, 40)
The resulting coordinates of the transformed polygon are:
A''', B''', C''', D''' = (0, 0), (80, 0), (80, 40), (0, 40)
Therefore, the transformation matrix to double the size of the polygon with point A located at the same place is:
[2 0 0]
[0 2 0]
[0 0 1]
Note: The third column of the transformation matrix remains unchanged, as it represents the translation component, and in this case, we didn't perform any translation.
Explanation: