Write a sequence of transformations that maps triangle ABC onto triangle A''B''C''.
ABC
A = 1,9
B = 3,12
C = 4,4
A''B''C''
A" = 3,-3
B'' = 5,-6,
C'' = 6,2.
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To determine the sequence of transformations that maps triangle ABC onto triangle A''B''C'', we need to identify the specific transformations that would result in the desired change in position and orientation.
1. Translation:
First, we can determine the required translation by finding the difference in coordinates between A and A'':
Δx = x'' - x = 3 - 1 = 2
Δy = y'' - y = -3 - 9 = -12
Therefore, we need to translate triangle ABC by +2 units in the x-direction and -12 units in the y-direction.
2. Scaling:
Next, let's identify the scaling factor to be used based on the difference in sizes between the original triangle ABC and the new triangle A''B''C''. Since the coordinates of A and A'' have changed, we need to determine the scale factor using the distance formula:
AB = √((x2 - x1)^2 + (y2 - y1)^2) = √((3 - 1)^2 + (12 - 9)^2) = √20
A''B'' = √((5 - 3)^2 + (-6 - (-3))^2) = √10
Scale factor = AB / A''B'' = √20 / √10 = √2
Therefore, we need to scale the translated triangle by a factor of √2.
3. Reflection:
Finally, we need to determine if any reflection is required to match the orientations of triangle ABC and A''B''C''. By comparing the slopes of the lines containing AB and A''B'':
Slope of AB = (12 - 9) / (3 - 1) = 3/2
Slope of A''B'' = (-6 - (-3)) / (5 - 3) = -3/2
The slopes are equal in magnitude but opposite in sign, indicating that a reflection across the x-axis is necessary.
Summary of transformations:
1. Translate triangle ABC by +2 units in the x-direction and -12 units in the y-direction.
2. Scale the translated triangle by a factor of √2.
3. Reflect the scaled triangle across the x-axis.
By performing these three transformations in the specified order, you will be able to map triangle ABC onto triangle A''B''C''.