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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.

# Isometries: Level 3 Challenges

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).

The image is not drawn to scale.

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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