To describe a series of transformations that Lyndsey can use to move the old figure to the new figure, we first need to identify the transformations involved. Common transformations that maintain similarity between figures include translations, rotations, reflections, and dilations.
Series of Transformations:
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Translation: If the new figure is positioned differently but maintains the same shape and size as the original figure, Lyndsey can translate (or slide) the old figure to a new position. For example, if the old figure is at coordinates (x, y) and the new figure is at (x + a, y + b), then the transformation can be described as: \[ T(x, y) = (x + a, y + b) \]
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Rotation: If the new figure is rotated around a specific point compared to the old figure, then the transformation can involve a rotation. For instance, if she needs to rotate around the origin by an angle θ, the points (x, y) would be transformed to: \[ R_{\theta}(x, y) = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \]
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Reflection: If the new figure is a mirror image of the old figure, then a reflection transformation would be applied. For example, reflecting across the x-axis would result in: \[ M(x, y) = (x, -y) \]
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Dilation: If the figures are of different sizes but still similar, a dilation transformation may be used. This involves a scale factor k, where (x, y) would transform to: \[ D(x, y) = (kx, ky) \]
Explanation of Similarity:
The figures are similar if:
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Same Shape: Both figures have corresponding angles that are equal. This means that the internal angles of the figures match up perfectly.
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Proportional Sides: The lengths of corresponding sides are in proportion. For example, if the old figure has side lengths of a, b, c, and the new figure has side lengths of ka, kb, kc for some constant k, then the figures are similar.
If Lyndsey applies one or more of the transformations described above (depending on the specific changes made to the figure), it retains the properties mentioned, thus establishing that the figures are similar.
In summary, to determine the exact transformations needed, one would require the specific details or coordinates of both the old and new figures. However, the types of transformations noted can generally help Lyndsey successfully adjust the figure while ensuring that it remains similar to the original.