# Solution to the Semi-Circle and Quarter Circle in a Square Puzzle +-- {.image} [[SemiCircleandQuarterCircleinaSquare.png:pic]] > A semicircle and quarter circle inside a square. How long is the black tangent line? =-- ## Solution by [[Symmetry]] and [[Angle Between a Radius and Tangent]] +-- {.image} [[SemiCircleandQuarterCircleinaSquareLabelled.png:pic]] =-- In the above diagram, $G$ is the centre of the semi-circle. The line $B G$ crosses $E F$ at [[right-angles]] because $B H$ is a radius of the quarter circle and $E F$ is tangent to it, and the [[angle between a radius and tangent]] is $90^\circ$. Therefore $E F$ and $B G$ are perpendicular line segments that both cross the square $A B C D$. By [[symmetry]] they have the same length. Since $B G$ is made up of two radii (one from each circle), it has length $8 + 5 = 13$. Thus also $E F$ has length $13$.