Math  /  Geometry

QuestionWhich transformation carries the square below onto itself?
Answer a reflection over the yy-axis a reflection over the line y=x+3y=-x+3 a reflection over the line x=2x=2 a reflection over the line y=x+7y=x+7

Studdy Solution

STEP 1

1. The square is centered at the origin.
2. The sides of the square are parallel to the axes.
3. The square is symmetric about both the x-axis and y-axis.

STEP 2

1. Understand the properties of the square and its symmetries.
2. Analyze each reflection option to determine if it maps the square onto itself.
3. Identify the correct transformation.

STEP 3

Understand the properties of the square: - The square is symmetric about the x-axis and y-axis. - It is also symmetric about the origin, meaning it can be rotated or reflected across lines through the origin to map onto itself.

STEP 4

Analyze each reflection option:
- **Reflection over the y y -axis**: - This reflection will map the square onto itself because the square is symmetric about the y y -axis.
- **Reflection over the line y=x+3 y = -x + 3 **: - This line does not pass through the origin, and reflecting over it will not map the square onto itself.
- **Reflection over the line x=2 x = 2 **: - This line is vertical and does not pass through the origin, so reflecting over it will not map the square onto itself.
- **Reflection over the line y=x+7 y = x + 7 **: - This line is diagonal and does not pass through the origin, so reflecting over it will not map the square onto itself.

STEP 5

Identify the correct transformation: - The only reflection that maps the square onto itself is the reflection over the y y -axis.
The transformation that carries the square onto itself is:
a reflection over the y-axis \boxed{\text{a reflection over the } y\text{-axis}}

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