Think about it ...
Julie says that a rotation of 180° around the origin is the same as the reflection in the x-axis followed by a reflection in the y-axis. Is Julie correct? Justify your answer and be sure to include a diagram.
I just wanna know does Julie correct or not. And why.
start with general point (x,y)
reflection in the x-axis ----> (x, -y)
then a reflection in the y-axis ---->(-x,-y)
reflection in the origin: (x,y) ----> (-x,-y)
Yup, Julie is correct.
Thank you very much ^^
To determine whether Julie is correct or not, let's analyze the situation step by step.
First, we need to understand what a rotation of 180° around the origin means. In a two-dimensional coordinate plane, the origin is the point (0, 0), where the x-axis and the y-axis intersect. A rotation of 180° around the origin means that every point is rotated by 180° in a counterclockwise or clockwise direction (since the origin remains fixed).
Now, let's consider what happens when we reflect a point in the x-axis. This reflection essentially flips the point over the x-axis. So, if a point is originally located at (x, y), after reflecting it in the x-axis, it will be located at (x, -y).
Next, let's examine the effect of reflecting a point in the y-axis. This reflection flips the point over the y-axis. Therefore, if a point is located at (x, y), after reflecting it in the y-axis, it will be located at (-x, y).
Now, let's compare the final location of a point after a rotation of 180° with the final location after a reflection in the x-axis followed by a reflection in the y-axis.
Suppose we have a point P at coordinates (x, y). After a rotation of 180° around the origin, P will be located at (-x, -y) because it is rotated by 180°.
On the other hand, if we reflect point P first in the x-axis, we get the new point P' at coordinates (x, -y). Then, if we reflect P' in the y-axis, we get the final point P'' at coordinates (-x, -y).
Comparing the final locations, we see that after the rotation, we have (-x, -y), whereas after the reflection in the x-axis followed by the reflection in the y-axis, we also have (-x, -y).
Since both methods produce the same result, it can be concluded that Julie is correct. A rotation of 180° around the origin is indeed equivalent to a reflection in the x-axis followed by a reflection in the y-axis.
I'm sorry, but as an AI text-based bot, I cannot provide a diagram. However, you can visualize this by drawing a coordinate plane and plotting a few points. Then, you can try rotating them 180° around the origin and compare the final locations with the result of reflecting them in the x-axis followed by the y-axis. This visual representation will reinforce the conclusion.