We received quite a few correct solutions for this month's puzzle. In case it defeated you, the key was to look for symmetries in the information you are given, and to use

*similar triangles.*

## Part I

With $y$ and $z$ as in the diagram, $x=z+1$, so it is sufficient to find $z$. By similar triangles, $z:1$ as $1:y$, so $zy=1.$

By Pythagoras' Theorem, \begin{eqnarray*} (z+1)^2 + (y+1)^2 &= 16,\\ z^2 + 2z + 1 + y^2 +2y +1 &= 16,\\ (z+y)^2 - 2zy + 2z + 2y + 2 &= 16. \end{eqnarray*} But $zy=1$, so $$(z+y)^2 + 2(z + y) - 16 =0.$$ Solving this equation for $z+y$, and taking only the positive solution, yields $$z+y = \sqrt{17}-1.$$ Now we find $z-y$. $$ (z+y)^2 - (z-y)^2 = 4zy = 4,$$ \begin{eqnarray*} (z-y)^2 &=& (z+y)^2 - 4\\ &=& (\sqrt{17}-1)^2 - 4\\ &=&17 -2\sqrt{17} + 1 - 4\\ &=& 14 - 2\sqrt{17}. \end{eqnarray*} So $$z+y = \sqrt{14 - 2\sqrt{17}}$$ and \begin{eqnarray*} 2z &=& (z+y) - (z-y)\\ &=& (\sqrt{17}-1) - (\sqrt{14 - 2\sqrt{17}}) \\ &=& 5.52. \end{eqnarray*} Therefore $$x = z+1 = 2.76 + 1 = 3.76.$$

## Part II

There are a number of approaches to this problem, but the one presented here follows from the observation that the information given is left-right symmetric - in other words, finding the two vertical heights will be equally easy (or hard!). We will label these two heights $y$ and $z$, as in the diagram.

By similar triangles, and dividing $x$ up into two segments $c$ and $d$, we see that $y:x$ as $8:d$ and $z:x$ as $8:c$; therefore $$z = 8x/c \mbox{ and} y=8x/d.$$ Adding gives $y+z = 8x^2/cd$ and multiplying gives $yz = 64x^2/cd.$ Combining these two results gives \begin{eqnarray*} yz &=& 8(y+z);\\ (y+z)/yz &=& 1/8;\\ 1/z + 1/y &=& 1/8. \end{eqnarray*} Leaving this
equation to one side for a moment, we can use Pythagoras' Theorem to find each of $y$ and $z$ in terms of our unknown $x$, as shown in the diagram above. Combining these two expressions gives that $z^2 - y^2 = 500$.

Using Pythagoras' Theorem, we draw a triangle to represent this relationship:

On its own, the information in this diagram is not enough to solve the triangle - for that, we would need one more bit of information. But now we can use the identity $yz = 8(y+z).$ All we have to do is find some combination of trig functions of $A$ that takes the form $1/z + 1/y$. Since $cos(A) = \sqrt{500}/y$ and $tan(A) = y/\sqrt{500}$, it is easy to see that $$ cos(A) + 1/tan(A) = \sqrt{500}(1/z + 1/y) = \sqrt{500}/8 = 2.7951.$$ Using a calculator gives $A = 27^o38^{\prime}30^{\prime\prime}$ and $z= \sqrt{500}/cos A = 25.24$. Therefore $$x= \sqrt{900 - (25.24)^2} = 16.2.$$