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Isometries

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.

Translation

Point \((2, 1)\) is moved to point \((10,11)\) by \[T: (x, y) \rightarrow (x+a, y+b).\] What is \(a+b\)?

If we translate the point \((9, 3)\) by \((13, 2),\) what would be the new point?

The curve \(y=2x^2+3x-9\) undergoes a parallel translation by \(7\) units in the positive direction of the \(x\)-axis and \(-3\) units in the positive direction of the \(y\)-axis. If the equation of the resulting curve is \(y=ax^2+bx+c,\) what is \(a+b+c?\)

Consider a translation of axes \[T: (x, y) \to (x+a, y+b)\] such that the coordinates of a point \(P=(7, 4)\) with respect to the new system of coordinates after translation are \(P'=(10,-11).\) What is the value of \(a+b?\)

If the point \((m,n)\) is shifted by \(6\) in the positive \(x\) direction and by \(-12\) in the positive \(y\) direction, then it coincides with the center of the circle \[x^2+y^2-6x+2y-13=0.\] What is the value of \(m+n?\)

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