Which of the following would be a line of reflection that would map ABCD onto itself?
Answer (2)
Identify the coordinates of the vertices of the rectangle ABCD.
Determine the midpoints of the sides and diagonals to find potential lines of symmetry.
Verify that the lines x=0 and y=1.5 map the rectangle onto itself.
Conclude that the lines of reflection are x=0 and y=1.5.
Explanation
Problem Analysis The problem asks us to identify a line of reflection that maps the rectangle ABCD onto itself. From the grid, we can identify the coordinates of the vertices as A(1,0), B(1,3), C(-1,3), and D(-1,0). A line of reflection that maps a shape onto itself is a line of symmetry. A rectangle has two lines of symmetry: one that runs horizontally through the middle and one that runs vertically through the middle.
Finding the Vertical Line of Symmetry Let's find the equation of the vertical line of symmetry. This line passes through the midpoints of AD and BC. The midpoint of AD is (0,0) and the midpoint of BC is (0,3). Thus, the vertical line of symmetry is x=0.
Finding the Horizontal Line of Symmetry Now, let's find the equation of the horizontal line of symmetry. This line passes through the midpoints of AB and CD. The midpoint of AB is (1, 1.5) and the midpoint of CD is (-1, 1.5). Thus, the horizontal line of symmetry is y=1.5.
Verification of x=0 The line x=0 maps A(1,0) to D(-1,0), B(1,3) to C(-1,3), C(-1,3) to B(1,3), and D(-1,0) to A(1,0). Thus, x=0 is a line of reflection that maps ABCD onto itself.
Verification of y=1.5 The line y=1.5 maps A(1,0) to (1,3)=B, B(1,3) to (1,0)=A, C(-1,3) to (-1,0)=D, and D(-1,0) to (-1,3)=C. Thus, y=1.5 is a line of reflection that maps ABCD onto itself.
Conclusion Therefore, the lines of reflection that map rectangle ABCD onto itself are x=0 and y=1.5.
Examples
Lines of symmetry are useful in architecture and design. For example, when designing a building, architects often use lines of symmetry to create a balanced and aesthetically pleasing structure. Similarly, in graphic design, lines of symmetry can be used to create logos and other visual elements that are both visually appealing and easy to recognize. Understanding lines of symmetry helps ensure that designs are balanced and harmonious.
The lines of reflection that map rectangle ABCD onto itself are x=0 and y=1.5. These lines correspond to the vertical and horizontal midpoints of the rectangle, ensuring symmetry. By reflecting across these lines, the rectangle maintains its shape and position.
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