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In the diagram, the letter S is made from two arcs, $KL$ and $MN,$ which are each $\frac{5}{8}$ of the circumference of a circle with radius 1, and the line segment $\overline{LM},$ which is tangent to both circles. Also, the tangent to the top circle at $K$ touches the bottom circle, and the tangent to the bottom circle at $N$ touches the top circle. What is the length of $\overline{LM}$? enter image description here

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2 Answers 2

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Draw a line from the center $C$ of the upper circle to the center of the second circle. That line will intersect ${LM}$ in the middle in some point $X$. The triangle $XCL$ has a right angle at $L$ (why?) and two sides of equal length, because the angle in $C$ is $\Pi/4$. So ${LX}$ has the same length as ${CL}$ which has length $1$ (the radius).

So the length of ${LM}$ is $=2$

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If you take the centre of the upper circle as the origin ($o$) of a rectangular coordinate system, and $K$ as the point $(1,0)$, then $\displaystyle L=\left(\frac{-1}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right)$. The slope of $LM$ is $-1$.

If $LM$ meets the $y$-axis at $P$. Then $OL=LP=1$. By symmetry, $LM=2$.

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