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 straight cut, that is perpendicular to this surface of the brownie, 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 ABCABC is such that AC=24AC=24cm, CB=10CB=10cm and ACB=90 \angle ACB=90^{\circ}. A point DD on CBCB is such that when point CC is reflected upon line ADAD, the reflected point will be on line ABAB. The length of CD=abCD=\frac{a}{b} where aa and bb are coprime integers. Find the value of a+ba+b.

The angle between two mirrors is θ \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 θ \theta?

Consider a parabolic mirror with vertex OO. A light ray originating from focus SS of the parabola, strikes the mirror at a point PP such that it gets deviated by 3030^{\circ} after reflection.

Now, if we draw a tangent at point PP to the same parabola, it intersects the tangent at vertex at point QQ. Find OSQ\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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