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Symmetries are one of the most beautiful part of Geometry. Slide into reflections, rotations, translations, and more, and you'll be on your way to geometric artistry.



The lengths of the lines in the above figure (a half house) are as follows: \[\lvert \overline{AB} \rvert = 9, \lvert \overline{BC} \rvert = 3,\]\[ \lvert \overline{CD} \rvert = 3, \lvert \overline{DE} \rvert = 3, \lvert \overline{EA} \rvert = 12.\] If the shape is flipped across the line segment \(\overline{AE}\) to complete a house, what will be the perimeter of the house?

The reflection of the circle \(x^2+y^2-20x-20y+196=0\) in the origin \((0,0)\ \) is \(\ x^2+y^2+ax+by+c=0.\) What is \(a+b+c\)?

The point \(A=(-4,5)\) is reflected in the line \(y=x.\) What are the coordinates of the resulting image?

If rectangle \(ABCD\) in the above graph is reflected in the line \(y = -1,\) what is the image of point \(D?\) The side length of each square in the graph paper is \(1.\)

In the above diagram, shape \(B\) on the right is the reflection of shape \(A\) in the red line. Given the following numbers: \[L = 48, \quad N = 3, \quad M = 6\] in the diagram, what is the value of \(a - b + c - d?\)


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