Finding the Radius of a Tangent Circumference in a Right Triangle

[Gjmdp]

Homework Statement

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.

Homework 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).

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 :)[/B]

 

[ehild]

Homework Statement

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.

Homework 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).

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 :)[/B]

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

 

[Gjmdp]

Did you mean a circle with centre M and radius r? What did you denote by x?

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

 

[ehild]

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

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.

 

[Gjmdp]

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.
View attachment 106239
You know the tangent of the angle x in the blue right triangle, and also tan(2x) from the triangle ABC.

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

 

[haruspex]

Then, tan(x)=12/r and tan(2x)=12/5; Am I right

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

 

[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.

 

[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?

 

[haruspex]

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?

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.

 

[Gjmdp]

Thank you very much! :)