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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.

Level 3

I came back from letting my brownies cool to find that some crazy individual had cut themselves a huge rectangular piece out of the middle of my brownie. The piece does not even align with the tray. Is it still possible to use a single vertical cut to cut the remaining brownie area into two pieces of equal area?

If not, prove why not. If it's possible, how can you do it?

A right angle triangle \(ABC\) is such that \(AC=24\)cm, \(CB=10\)cm and \( \angle ACB=90^{\circ}\). A point \(D\) on \(CB\) is such that when point \(C\) is reflected upon line \(AD\), the reflected point will be on line \(AB\). The length of \(CD=\frac{a}{b}\) where \(a\) and \(b\) are coprime integers. Find the value of \(a+b\).

The angle between two mirrors is \( \theta\). If the incident ray that is parallel to mirror 1 is reflected 5 times in the setup, and becomes parallel to mirror 1 again. What is the measure (in degrees) of \( \theta\)?

Consider a parabolic mirror with vertex \(O\). A light ray originating from focus \(S\) of the parabola, strikes the mirror at a point \(P\) such that it gets deviated by \(30^{\circ}\) after reflection.

Now, if we draw a tangent at point \(P\) to the same parabola, it intersects the tangent at vertex at point \(Q\). Find \(\angle OSQ\) (in degrees).

How many lines of symmetry does a parallelogram possess?

Note: The parallelogram has no special properties as of a rectangle or a rhombus.

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