For a square drawn in the plane, name the identity symmetry and state what it does to every point of the square.

In the composition \(T\circ U\), which transformation is applied first?

An equilateral triangle is rotated by \(120^\circ\). Is this a symmetry of the unlabelled triangle? Explain briefly.

A rectangle that is not a square is rotated by \(90^\circ\) about its centre. Is this a symmetry of the rectangle?

For the rotational symmetries \(r_0,r_{120},r_{240}\) of an equilateral triangle, compute \(r_{120}\circ r_{240}\).

Find the inverse of the \(240^\circ\) rotation symmetry of an equilateral triangle.

A regular pentagon is rotated by \(144^\circ\), then by \(216^\circ\). Which rotation symmetry is the composition equivalent to?

A transformation preserves the area \(A\) of a triangle but changes its shape. Is area preservation alone enough to make it a symmetry of the triangle as a geometric shape?

The rotations of a square are \(r_0,r_{90},r_{180},r_{270}\). Compute \(r_{270}\circ r_{180}\) and express the answer as one of these rotations.

A circular potential \(V(x,y)=x^2+y^2\) is transformed by rotating the coordinates through any angle about the origin. Explain why this rotation is a symmetry of the potential.

Let \(T\) and \(U\) be reversible symmetries of the same object. Explain why \(T\circ U\) is also a symmetry.

For rotations of a regular hexagon, let \(r\) be rotation by \(60^\circ\). Compute \(r^4\circ r^5\) and justify the reduction.

A square has vertices labelled \(1,2,3,4\) clockwise. A \(90^\circ\) rotation preserves the unlabelled square but moves the labels. Is it a symmetry of the labelled square? Explain.

A shape has exactly two known symmetries: the identity \(e\) and a reflection \(s\). Use composition to determine \(s^{-1}\).

A transformation \(T\) of a finite object is a symmetry, and applying \(T\) three times gives the identity. Show that \(T^{-1}=T^2\).

A pattern on a line is unchanged by translation by \(2\) metres. Explain why translation by \(-2\) metres is also a symmetry.

A wallpaper pattern is unchanged by translations \(A\) and \(B\). Give a careful reason why \(A\circ B^{-1}\) is also a symmetry.

A student says, "Every length-preserving transformation of a labelled triangle is a symmetry." Diagnose the error.

Two symmetries \(S\) and \(T\) of an object satisfy \(S\circ T\ne T\circ S\). Does this contradict the claim that symmetries can be composed? Explain.

A physical system is described by two quantities: its circular shape and a marked direction arrow on it. Which rotations are symmetries, and why might this differ from a plain circle?