State the theorem of Pythagoras concerning the three sides of a right-angled triangle.

A diagram showing the triangle $ABC$ and the semicircles $S_1$ and $S_2$.

The diagram shows an isosceles triangle \(ABC\) right-angled at \(B\). Semicircles \(S_1\) and \(S_2\) are drawn on \(AB\) and \(AC\) as diameter, as shown.

  1. Explain why \(S_2\) passes through \(B\).
  2. If \(AB = \quantity{2x}{cm}\), show that the area shaded vertically is \(\quantity{\tfrac{1}{2}(\pi - 2)x^2}{cm^2}\).
  3. Prove that the area shaded horizontally is half the area of the triangle \(ABC\).