Square inside a triangle problem [see desc.]

Question

I Have a problem solving this problem only using high school math(Geometry) without using digital help (computer). The solution should be $\cfrac{11\sqrt{77}}{2}$.

How can I get this solution by hand? I get only a fourthgrade equation that I can not solve: $(x+1)^2(x^2+1)= 100x^2$ using similar triangles. There most be an easier way?

Answer 1

It should be $$\frac{x}{x+1}=\frac{\sqrt{x^2+1}}{10},$$ which indeed gives $$100x^2=(x+1)^2(x^2+1).$$

Now, let $x+\frac{1}{x}=t$.

Thus, $$100=(t+2)t$$ or $$(t+1)^2=101,$$ which gives $$t=\sqrt{101}-1$$ and $$x+\frac{1}{x}=\sqrt{101}-1.$$

Answer 2

Call the upper part of the vertical leg $y$. Then $$(1+x)^2+(1+y)^2=100$$ and ${y\over1}={1\over x}$, hence $xy=1$. Letting $x+y=:s$ we therefore have $$2+2s+(s^2-2xy)=100\ ,$$ or $(s+1)^2=101$. This gives $s=\sqrt{101}-1$, so that by Vieta's theorem $x$ and $y$ are the solutions of the quadratic $$z^2-(\sqrt{101}-1)z+1=0\ .$$