# Solution to the Three Congruent Triangles in a Semi-Circle Puzzle +-- {.image} [[ThreeCongruentTrianglesinaSemiCircle.png:pic]] > The three triangles are congruent. What’s the area of the semicircle? =-- ## Solution by [[Circle Geometry]] +-- {.image} [[ThreeCongruentTrianglesinaSemiCircleLabelled.png:pic]] =-- In the diagram above, the point $O$ is on the base of the semi-circle so that $O C$ is [[parallel]] to $A B$. Since $A O$ is parallel to $B C$, the [[quadrilateral]] $A O C B$ is a [[parallelogram]]. The lengths of $A B$ and $B C$ are the same since the triangles are all congruent, so the quadrilateral is actually a [[rhombus]]. This means that the lengths of $O A$ and $O C$ are equal (and both equal to the length of $B C$). The point $O$ is therefore on the diameter and equidistant from two points on the circumference, hence is the centre of the circle. The radius is therefore $10$ and the area of the semi-circle is $50 \pi$.