2
$\begingroup$

Hi I got a 10 cm long line, and it touches point 1,1

I need to calculate where it touches x and y.

enter image description here

If I think of it like an triangle i get the following information.

  • One side is 10 cm.
  • You get an angle of 90
  • and an Height of 1 cm.

But how do i calculate the rest?

UPDATE Figured out that its know as the Ladder problem. http://www.mathematische-basteleien.de/ladder.htm

I also updated the image to make it more clear.

  • 0
    @Also the link doesn't contain a general solution, only a specific solution for length $5$. It would be good to be able to compare your answer with Fabian's.2011-04-10

2 Answers 2

2

Looking at your figure, I do not think any of the height is $1$.

There are similar triangles in your figure: the large triangle with hypotenuse $10$ and catheti $x$ and $y$ is similar to the triangle with catheti $1$ and $y-1$ and also similar to the one with catheti $x-1$ and $1$. Using this we get that $\frac{x}{y} = x-1.$ Additionally, we know that $x^2 +y^2 =10^2$. Plugging in the relation $y= x/(x-1)$, we obtain $x^2 + \left( \frac{x}{x-1}\right)^2 = 10^2$ which is equivalent to $x^2 + x^2 (x-1)^2 = 100 (x-1)^2$ with the (only positive) solution (up to exchanging $x$ and $y$) $x= \frac{1}{2} \left[\sqrt{101} +1 - \sqrt{2 (49- \sqrt{101})}\right]\approx 1.11$ and $y=\frac{1}{2} \left[\sqrt{101} +1 +\sqrt{2 (49- \sqrt{101})}\right] \approx 9.94.$

  • 0
    Wow, I've never seen the word cathetus (plural catheti) used for a side of a right triangle other than the hypotenuse. I need to get out more. http://en.wikipedia.org/wiki/Cathetus2011-04-11
3

(if I understand correctly)

the sides of the triangle are $a,b,c$, with $c=10$. If you leave out the square, you get two small triangles which are similar. Hence $(a-1)/1=1/(b-1)$, i.e. $(a-1)(b-1)=1$, or $ab=a+b$. We also know $a^2+b^2=c^2$. From here you get $(a+b-1)^2=c^2+1$. So $(a-1)+(b-1)=-1+\sqrt{c^2+1}$, $(a-1)(b-1)=1$, i.e. $a-1$ and $b-1$ are the solutions of $x^2+(1-\sqrt{c^2+1})x+1=0$.

  • 0
    undeleted at yoriki's request :) (sorry for the $b$ notation - I had it before it appeared on the picture)2011-04-10