Question # 17
Solution:
Graphing the vertices of a triangle and transforming by first translating
(x, y) → (x -3, y - 3) and then reflecting in the line y = x.
As A(-1, 1), B(0, 3) and C(3, 4)
After Translation: (x, y) → (x - 3, y - 3)
A(-1, 1) → A'(-4, -2)
B(0, 3) → B'(-3, 0)
C(3, 4) → C'(0, 1)
After Reflection: in the line y = x
The reflection of the point (x, y) across the
line y = x is the point (y, x).
A'(-4, -2) → A''(-2, -4)
B'(-3, 0) → B''(0, -3)
C'(0, 1) → C''(1, 0)
Pleas check the top right side of the attached figure a.
Is the image same if the order of the transformation is switched ? (Reflection, then glide). Changing the order of transformation.
A(-1, 1), B(0, 3) and C(3, 4)
After Reflection: in the line y = x
The reflection of the point (x, y) across the line y = x is the point (y, x).
A(-1, 1) → A'(1, -1)
B(0, 3) → B'(3, 0)
C(3, 4) → C'(4, 3)
After Translation: (x, y) → (x - 3, y - 3)
A'(1, -1) → A''(-2, -4)
B'(3, 0) → B''(0, -3)
C'(4, 3) → C''(1, 0)
No, the image is no longer the same if order of the transformation is switched as shown at the right bottom of figure a.
Answering the YES and NO Questions:
Is the composition a glide reflection?
No, the composition is not a glide reflection as glide reflection is commutative. If we reverse the direction of the composition, the outcome will be affected.
Check the figure at top right side of figure a, and compare it with the figure at the bottom right side of figure a. They are not the same.
Is the image same if the order of the transformation is switched ? (Reflection, then glide)
No, the image is no longer the same if order of the transformation is switched as shown at the right bottom figure of figure a.
The top right side of figure a was transformed after first translation of (x, y) → (x - 3, y - 3) and then doing reflection across the line y = x. While the bottom right side of figure a was made by first reflection across the line y= x and then translation of (x, y) → (x - 3, y - 3. So, after changing the order both figures do not remain the same.
Question # 18
Solution:
Graphing the vertices of a triangle and transforming by first 90 rotating clockwise and then reflection in the y axis.
A(-1, 1), B(0, 3) and C(3, 4)
After transforming by first 90 rotating clockwise about the origin. The new position of point (x, y) will be (y, -x)
A(-1, 1) → A'(1, 1)
B(0, 3) → B'(3, 0)
C(3, 4) → C'(4, -3)
After Reflection: in the line y axis
The reflection of the point (x, y) in the line line y-axis is the point (-x, y).
A'(1, 1) → A''(-1, 1)
B'(3, 0) → B''(-3, 0)
C'(4, -3) → C''(-4, -3)
Please check the top right side of figure b.
If the order of the transformation were performed in reverse order ? (Reflection, then rotation), would the final image by the same?
A(-1, 1), B(0, 3) and C(3, 4)
After Reflection: in the line y-axis
The reflection of the point (x, y) in the line y-axis is the point (-x, y).
A(-1, 1) → A'(1, 1)
B(0, 3) → B'(0, 3)
C(3, 4) → C'(-3, 4)
After transforming by first 90 rotating clockwise about the origin. The new position of point (x, y) will be (y, -x)
A'(1, 1) → A''(1, -1)
B'(0, 3) → B''(3, 0)
C'(-3, 4) → C''(4, 3)
No, the image is no longer the same if order of the transformation is switched as shown at the right bottom of figure b.
Answering the YES and NO Questions:
If the transformation were performed in reverse order (reflection. then rotation), would the final image be the same?
No, if we reverse the direction of the composition, the final image would not be same.
Check the figure at top right side of figure b, and compare it with the at the bottom right side of figure b. They are not the same.
Would the reflection of triangle ABC in the line y = x have produced the same final image?
Yes, the reflection of triangle ABC in the line y = x would have produced the same final image. Compare the figure of the bottom left with the bottom right of figure b.
As A(-1, 1), B(0, 3) and C(3, 4)
After Reflection: in the line y = x
The reflection of the point (x, y) across the
line y = x is the point (y, x).
A(-1, 1) → A''(1, -1)
B(0, 3) → B''(3, 0)
C(3, 4) → C''(4, 3)
Hence, the reflection of triangle ABC in the line y = x would have produced the same final image. Compare the figure of the bottom left with the bottom right of figure b.
Keywords: transformation, reflection, rotation, translation, triangle
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