Symmetry

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Plot the graph of x^y = y^x in the first quadrant and explain its properties.

When a strip has vertical symmetry there always seems to be a second place where a mirror line could go. Perhaps you can find a design that has only one mirror line across it. Or, if you thought that was impossible, could you explain why ?

Find the shape and symmetries of the two pieces of this cut cube.

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Points off a rolling wheel make traces. What makes those traces have symmetry?

I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only one centre of rotation ? Or if you thought that was impossible, could you say why ?