For triangle ABC , which transformation composition is Commutative? a) rotate 30 degrees and then translate 2 units down b) translate 5 units to the right and rotate 90 degrees c) reflect across the y-axis and then rotate 90 degrees d) reflect across the y -axis and then reflect across the x-axis
I think it's D but it's just a guess....thanks!
D it is
When you combine translations all bets are off.
The composition of transformations is commutative if the order in which the transformations are applied does not affect the final result.
Let's consider each option and see if the composition is commutative:
a) Rotate 30 degrees and then translate 2 units down.
b) Translate 5 units to the right and rotate 90 degrees.
c) Reflect across the y-axis and then rotate 90 degrees.
d) Reflect across the y-axis and then reflect across the x-axis.
For option a), if we first rotate the triangle 30 degrees and then translate it 2 units down, the final triangle will be different from the result if we apply the transformations in the reverse order. Therefore, option a) is not commutative.
For option b), if we first translate the triangle 5 units to the right and then rotate it 90 degrees, the final triangle will be the same as if we apply the transformations in the reverse order. Therefore, option b) is commutative.
For option c), if we first reflect the triangle across the y-axis and then rotate it 90 degrees, the final triangle will be the same as if we apply the transformations in the reverse order. Therefore, option c) is commutative.
For option d), if we first reflect the triangle across the y-axis and then reflect it across the x-axis, the final triangle will be different from the result if we apply the transformations in the reverse order. Therefore, option d) is not commutative.
Based on the analysis above, the correct answer is option b) "translate 5 units to the right and rotate 90 degrees", as it is the only composition of transformations that is commutative.
To determine which transformation composition is commutative, we need to check if changing the order of the transformations results in the same final outcome. Let's analyze each option:
a) Rotate 30 degrees and then translate 2 units down.
If we perform the rotation first, followed by the translation, the triangle will be rotated 30 degrees and then shifted 2 units down. The end result will be a different position compared to the alternative order of operations.
b) Translate 5 units to the right and rotate 90 degrees.
If we perform the translation first, followed by the rotation, the triangle will be shifted 5 units to the right and then rotated 90 degrees. Again, the final outcome will be different from the alternative order of operations.
c) Reflect across the y-axis and then rotate 90 degrees.
If we perform the reflection across the y-axis first, followed by the rotation, the triangle will be reflected and then rotated. Once again, the final outcome will be different if we change the order of operations.
d) Reflect across the y-axis and then reflect across the x-axis.
If we perform the reflection across the y-axis first, followed by the reflection across the x-axis, the triangle will be reflected twice. This means that changing the order of operations will not make a difference since reflecting twice will result in the same final outcome.
Therefore, the correct answer is d) Reflect across the y-axis and then reflect across the x-axis. This composition is commutative since changing the order of operations does not affect the final outcome.