# Triangle and circumference

1. Sep 19, 2016

### Gjmdp

1. The problem statement, all variables and given/known data
Let AC=5 and BC=12. In the triangle ABC, with angle C=90, point M is in AC. A circumference with center M and radius r is tangent to AB and tangent to BC in C. Set r.

2. Relevant equations
This should envolve basic trigonometry, and Thales' theorem; but not sure ( if I knew the equations for solving the problem, I would alredy knew the answer).

3. The attempt at a solution

By Thales' Therem: AC/BC=(AC-r)/x; then: 5/12=(5-r)/x. But I don't know how to get x. I've tried many proportions and no one just works. I also tried the 2 Thales' theorem, and didn't work either. Let me tell I know the answer, r=12/5, and that this is not for any class, just found on internet, but I can't get to know how to solve it. If any help, appreciate :)

2. Sep 19, 2016

### ehild

Did you mean a circle with centre M and radius r? What did you denote by x? Draw a picture of the problem.

3. Sep 19, 2016

### Gjmdp

Neither M or r are denoted by x. X is an unknow number, and I use it to make proportions with Thales'.

4. Sep 19, 2016

### ehild

What do you mean on "proportion with Thales"? Thales Theorem states that a triangle inscribed into a semicircle is a right triangle. http://mathworld.wolfram.com/ThalesTheorem.html
There is no inscribed triangle in the problem.
You should draw a figure to unterstand the problem.

You know the tangent of the angle x in the blue right triangle, and also tan(2x) from the triangle ABC.

5. Sep 19, 2016

### Gjmdp

Then, tan(x)=12/r and tan(2x)=12/5; Am I right?
But then, r does not equal 12/5, which is the solution to the problem

Last edited: Sep 19, 2016
6. Sep 19, 2016

### haruspex

No. tan is opposite divided by adjacent. You seem to have it backwards.

7. Sep 19, 2016

### LCKurtz

If you call $MA = s$ and where the green radius hits AB as point D you can use that triangle AMD is similar to triangle ABC and $r+s=5$. You don't need any trig.

8. Sep 20, 2016

### Gjmdp

OK guys, thank you very much, now I know how to solve the problem! One last question: why green radius makes 90 degrees with AB? How do you know that?

9. Sep 20, 2016

### haruspex

A radius of a circle to a point on its circumference makes a right angle to the tangent at the same point. It's more-or-less definition of a tangent. This generalises to smooth curves and instantaneous centres of arc.

10. Sep 24, 2016

### Gjmdp

Thank you very much!!! :)