Which transformations can be used to carry $A B C D$ onto itself? The point of rotation is $(3,2)$. Check all that apply.
A. Reflection across the line $y=2$
B. Reflection across the line $x=3$
C. Rotation of $90^{\circ}$
D. Translation two units down
Answer (1)
Reflection across the line y = 2 carries A BC D onto itself if A BC D is symmetric with respect to y = 2 .
Reflection across the line x = 3 carries A BC D onto itself if A BC D is symmetric with respect to x = 3 .
Rotation of 9 0 ∘ about ( 3 , 2 ) carries A BC D onto itself if A BC D has rotational symmetry of order 4 about ( 3 , 2 ) .
Translation two units down carries A BC D onto itself if A BC D has translational symmetry in the vertical direction with a period of 2 units.
Possible transformations are reflection across the line y = 2 , reflection across the line x = 3 , and rotation of 9 0 ∘ .
The transformations that can carry A BC D onto itself are A, B, and C. A , B , C
Explanation
Analyze the problem Let's analyze the given options to determine which transformations carry figure A BC D onto itself, given that the point of rotation is ( 3 , 2 ) . We need to determine the symmetry properties of the figure A BC D . Without a visual representation or further information about the figure A BC D , we must consider the general implications of each transformation.
Reflection across y=2 A. Reflection across the line y = 2 : This transformation would carry A BC D onto itself if the figure is symmetric with respect to the line y = 2 . This means that for every point ( x , y ) in A BC D , the point ( x , 4 − y ) must also be in A BC D .
Reflection across x=3 B. Reflection across the line x = 3 : This transformation would carry A BC D onto itself if the figure is symmetric with respect to the line x = 3 . This means that for every point ( x , y ) in A BC D , the point ( 6 − x , y ) must also be in A BC D .
Rotation of 90 degrees C. Rotation of 9 0 ∘ : A 9 0 ∘ rotation about the point ( 3 , 2 ) would carry A BC D onto itself if the figure has rotational symmetry of order 4 about the point ( 3 , 2 ) . This means that after rotating the figure by 9 0 ∘ , 18 0 ∘ , 27 0 ∘ about ( 3 , 2 ) , it looks the same.
Translation two units down D. Translation two units down: This transformation would carry A BC D onto itself if the figure has translational symmetry in the vertical direction with a period of 2 units. This means that for every point ( x , y ) in A BC D , the point ( x , y − 2 ) must also be in A BC D .
Determine possible transformations Without knowing the exact shape of A BC D , we cannot definitively say which transformations work. However, if A BC D is a square centered at ( 3 , 2 ) with sides parallel to the axes, then reflection across the lines y = 2 and x = 3 and a rotation of 9 0 ∘ about ( 3 , 2 ) would carry A BC D onto itself. A translation two units down would not carry A BC D onto itself unless the figure has specific translational symmetry. Therefore, options A, B, and C are possible.
Final Answer Based on the analysis, the transformations that can carry A BC D onto itself are reflection across the line y = 2 , reflection across the line x = 3 , and rotation of 9 0 ∘ .
Examples
Understanding symmetry and transformations is crucial in various fields, such as art, architecture, and design. For example, architects use reflections and rotations to create symmetrical building designs, ensuring balance and aesthetic appeal. Similarly, artists use these transformations to create patterns and tessellations in their artwork. In computer graphics, transformations are fundamental for manipulating objects in 3D space, allowing for realistic rendering and animation.
