Notes
semi-circle in a right-angled triangle solution

Solution to the Semi-Circle in a Right-Angled Triangle Puzzle

Solution by Angle Between a Radius and Tangent

With the points labelled as in the diagram above, the point labelled $O$ is the centre of the circle.

Angles $O \hat{A} C$ and $C \hat{B} O$ are both right-angles as they are the angle between a radius and tangent. Angle $A \hat{C} B$ is given as a right-angle. Therefore, quadrilateral $A O B C$ is a rectangle. The lengths of $O A$ and $O B$ are equal as they are radii of the semi-circle, so in fact $O A B C$ is a square. As the diameter of the semi-circle is $8$, the side length of the square is $4$ and its area is $16$. The triangle $A B C$ is half of that square and so has area $8$.