Rigid motions, congruence and similarity

Rotate the point $(2, 5)$ by $90^\circ$ anticlockwise about the origin. Where does it land?

Translate the point $(1, -1)$ by $(-3, 4)$. Where does it land?

A shape is reflected in the $x$-axis and then in the $y$-axis. Which single move has the same effect?

A triangle has area $5\ \mathrm{cm}^2$. It is dilated with scale factor $3$. What is the area of the image, in $\mathrm{cm}^2$?

Triangle $P$ has corners $(0, 0)$, $(3, 0)$, $(0, 4)$. Triangle $Q$ has corners $(0, 0)$, $(6, 0)$, $(0, 8)$. Which is true?

Which of these does every rigid motion keep the same? (Choose all that apply.)

Where does the point $(1, 1)$ go if you first translate by $(2, 0)$ and then rotate by $180^\circ$ about the origin?

Two mirror lines cross at a point $O$ at an angle of $30^\circ$. A shape is reflected in the first line and then in the second. The result is a rotation about $O$. By how many degrees?

In the Lean file, IsIsometry f means that f keeps all distances, and g ∘ f means “first f, then g”. Which statement says that the composite of two rigid motions is a rigid motion?

Points are pairs (x, y). Complete then(f, g) so that it returns a new function that applies f first and then g. Use it to check that reflecting in the $x$-axis and then in the line $y = x$ is the rotation by $90^\circ$, and that the other order is not.