Asked by dirt
Which quadrant will triangle X'Y'Z' be in if triangle XYZ begins in Quadrant IV and is reflected across the y-axis and then the x-axis?
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GPT-4o mini
To determine the final location of triangle X'Y'Z' after reflecting triangle XYZ across the y-axis and then the x-axis, we can analyze the transformations step by step.
1. **Initial Position in Quadrant IV**: In the Cartesian coordinate system, Quadrant IV is characterized by coordinates where \(x\) is positive and \(y\) is negative (i.e., any point in Quadrant IV has the form \((x, y)\) where \(x > 0\) and \(y < 0\)).
2. **Reflection Across the y-axis**: When a point \((x, y)\) is reflected across the y-axis, its new coordinates become \((-x, y)\). Therefore, if triangle XYZ has vertices in Quadrant IV, reflecting across the y-axis will move the triangle to Quadrant III since the new \(x\) coordinates will be negative while the \(y\) coordinates remain negative. So, after this step, triangle XYZ is in Quadrant III.
3. **Reflection Across the x-axis**: Now we reflect the triangle that is now in Quadrant III across the x-axis. When a point \((x, y)\) is reflected across the x-axis, its new coordinates become \((x, -y)\). In this case, since the triangle was in Quadrant III (where \(x < 0\) and \(y < 0\)), after reflecting across the x-axis, the new coordinates will be in Quadrant II. This is because the \(x\) coordinates remain negative while the \(y\) coordinates become positive.
Therefore, after performing both transformations, triangle X'Y'Z' will end up in **Quadrant II**.
1. **Initial Position in Quadrant IV**: In the Cartesian coordinate system, Quadrant IV is characterized by coordinates where \(x\) is positive and \(y\) is negative (i.e., any point in Quadrant IV has the form \((x, y)\) where \(x > 0\) and \(y < 0\)).
2. **Reflection Across the y-axis**: When a point \((x, y)\) is reflected across the y-axis, its new coordinates become \((-x, y)\). Therefore, if triangle XYZ has vertices in Quadrant IV, reflecting across the y-axis will move the triangle to Quadrant III since the new \(x\) coordinates will be negative while the \(y\) coordinates remain negative. So, after this step, triangle XYZ is in Quadrant III.
3. **Reflection Across the x-axis**: Now we reflect the triangle that is now in Quadrant III across the x-axis. When a point \((x, y)\) is reflected across the x-axis, its new coordinates become \((x, -y)\). In this case, since the triangle was in Quadrant III (where \(x < 0\) and \(y < 0\)), after reflecting across the x-axis, the new coordinates will be in Quadrant II. This is because the \(x\) coordinates remain negative while the \(y\) coordinates become positive.
Therefore, after performing both transformations, triangle X'Y'Z' will end up in **Quadrant II**.
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