Write a sequence of transformations that maps triangle ABC onto triangle A''B''C''.
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A in ABC is 1,9, B in ABC is 3,12, and C in ABC is 4,4.
A'' in A''B''C'' is 3,-3, B'' in A''B''C'' is 5,-6, and C'' in A''B''C'' is 6,2.
To map triangle ABC onto triangle A''B''C'', you can use a combination of translation, rotation, and reflection transformations. Here are the sequence of transformations:
1. Translation - Move triangle ABC so that point A coincides with point A'':
- Calculate the vector translation from A to A'':
(x'', y'') = (3, -3) - (1, 9) = (2, -12)
- Apply this translation to all three vertices of triangle ABC:
A' = A + (2, -12) = (1 + 2, 9 - 12) = (3, -3)
B' = B + (2, -12) = (3 + 2, 12 - 12) = (5, 0)
C' = C + (2, -12) = (4 + 2, 4 - 12) = (6, -8)
2. Rotation - Rotate triangle A'B'C' clockwise around a specific point to match the orientation of triangle A''B''C'':
- Determine the angle of rotation by comparing the orientations of A'B' and A''B'':
A'B' = (5 - 3, 0 - (-3)) = (2, 3)
A''B'' = (5 - 3, -6 - (-3)) = (2, -3)
The angle of rotation can be found using the dot product formula: cosθ = (A'B' ∙ A''B'') / (|A'B'| |A''B''|)
cosθ = (2 * 2 + 3 * (-3)) / (sqrt(2^2 + 3^2) * sqrt(2^2 + (-3)^2)) = (-5) / (sqrt(13) * sqrt(13)) = -5 / 13
θ = acos(-5 / 13)
- Choose a point as the center of rotation, which can be any of the vertices (A', B', or C'). Let's choose B' as the center.
- Rotate each vertex of triangle A'B'C' by angle θ around B':
A'' = Rotate(A', B', θ)
B'' = Rotate(B', B', θ) = B' (no change since the center of rotation remains fixed)
C'' = Rotate(C', B', θ)
3. Reflection - Reflect triangle A''B''C'' over a line to match the position of A''B''C'':
- Choose any line that passes through one of the vertices (A'', B'', or C''). Let's choose the line passing through A''.
- Reflect each vertex of triangle A''B''C'' over the chosen line:
A''' = Reflect(A'', A'')
B''' = Reflect(B'', A'')
C''' = Reflect(C'', A'')
The resulting vertices A''', B''', and C''' will form the transformed triangle A'''B'''C'''. The sequence of transformations mentioned above will map triangle ABC onto triangle A''B''C''.
To map triangle ABC onto triangle A''B''C'', we need to perform a series of transformations. Here are the step-by-step transformations:
1. Translation: Move triangle ABC so that point A coincides with point A''.
- Translate vector AA'': (3, -3) - (1, 9) = (2, -12)
- Apply this translation to all points of triangle ABC.
Result: Triangle ABC has been translated by vector (2, -12), and point A is now coinciding with point A''.
2. Scaling: Scale the translated triangle to match the size of triangle A''B''C''.
- Find the ratios of the corresponding side lengths:
- AB / A''B'' = sqrt((3-1)^2 + (12-9)^2) / sqrt((5-3)^2 + (-6+3)^2) = sqrt(13) / sqrt(25) = sqrt(13) / 5
- BC / B''C'' = sqrt((4-3)^2 + (4-12)^2) / sqrt((6-5)^2 + (2+6)^2) = sqrt(64) / sqrt(65) = 8 / sqrt(65)
- CA / C''A'' = sqrt((4-1)^2 + (4-9)^2) / sqrt((6-3)^2 + (2-(-3))^2) = sqrt(34) / sqrt(58) = sqrt(17) / sqrt(29)
- Multiply all coordinates of triangle ABC by the scaling factors:
- A' = (1, 9) * sqrt(13) / 5
- B' = (3, 12) * 8 / sqrt(65)
- C' = (4, 4) * sqrt(17) / sqrt(29)
Result: Triangle ABC has been scaled to match the size of triangle A''B''C''.
3. Translation: Move the scaled triangle to match the position of triangle A''B''C''.
- Translate vertex A' to A'':
- Translate vector A'A'': (3, -3) - (1 * sqrt(13) / 5, 9 * sqrt(13) / 5) = (3 - sqrt(13) / 5, -3 - 9 * sqrt(13) / 5)
- Apply this translation to all points of the scaled triangle ABC.
Result: Triangle ABC has been translated by the vector (3 - sqrt(13) / 5, -3 - 9 * sqrt(13) / 5), and it now matches the position and size of triangle A''B''C''.
Now triangle ABC is mapped onto triangle A''B''C'' using the sequence of transformations described above.