# Solution to the Two Semi-Circles Inside a Semi-Circle II Puzzle +-- {.image} [[TwoSemiCirclesInsideaSemiCircleII.png:pic]] > What's the diameter of the largest semicircle? =-- ## Solution by [[Angle Between a Radius and Tangent]], [[Pythagoras' Theorem]], [[Angle in a Semi-Circle]], and [[Similar Triangles]] +-- {.image} [[TwoSemiCirclesInsideaSemiCircleIILabelled.png:pic]] =-- With the points labelled as above, $B$ and $G$ are the centres of their respective circles and $D$ is directly below $F$. The length of $B G$ is $1 + 1.5 = 2.5$ and of $G C$ is $1.5$. As $A E$ is tangential to the yellow semi-circle at $C$, angle $G \hat{C} A$ is the [[angle between a radius and tangent]] so is a right-angle. Therefore, triangle $B C G$ is a [[right-angled triangle]]. Applying [[Pythagoras' theorem]] shows that $B C$ has length $2$. The full length of $A D$ is then $1 + 2 + 1.5 = 4.5$. Angle $A \hat{F} E$ is the [[angle in a semi-circle]] so is a right-angle. Therefore angles $A \hat{F} D$ and $D \hat{F} E$ add up to $90^\circ$ and so triangles $A D F$ and $F D E$ are [[similar]]. As $F D$ has length $1.5$ and $A D$ length $4.5$, the length of $D E$ is one third of that of $F D$ and so is $.5$. Therefore $A E$ has length $5$.