Finding the Radius of a Tangent Circumference in a Right Triangle

In summary, the conversation discusses a problem involving a triangle ABC with angles and sides given, and a circle with a center point M and radius r that is tangent to two sides of the triangle. The problem is to find the value of r. The conversation explores using Thales' theorem and basic trigonometry to solve the problem, and ultimately determines that it can be solved without trigonometry by using the similarity of two triangles. The conversation also discusses the definition of a tangent and how it relates to the problem.
  • #1
Gjmdp
148
5

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]
 
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  • #2
Gjmdp said:

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.
 
  • #3
ehild said:
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'.
 
  • #4
Gjmdp said:
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.
upload_2016-9-19_21-42-56.png

You know the tangent of the angle x in the blue right triangle, and also tan(2x) from the triangle ABC.
 
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  • #5
ehild said:
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
 
Last edited:
  • #6
Gjmdp said:
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.
 
  • #7
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
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
Gjmdp said:
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.
 
  • #10
Thank you very much! :)
 

Related to Finding the Radius of a Tangent Circumference in a Right Triangle

1. What is the formula for finding the circumference of a triangle?

The formula for finding the circumference of a triangle is c = a + b + c, where a, b, and c are the lengths of the three sides of the triangle.

2. Can a triangle have a circumference?

Yes, a triangle can have a circumference. The circumference of a triangle is the distance around the outside of the triangle, which is equal to the sum of the lengths of its three sides.

3. How do you find the circumference of an equilateral triangle?

The circumference of an equilateral triangle can be found by multiplying the length of one side by 3, since all three sides of an equilateral triangle are equal in length.

4. What is the relationship between the circumference and the perimeter of a triangle?

The circumference of a triangle refers to the distance around the outside of the triangle, while the perimeter refers to the total length of all its sides. For a triangle, the circumference is equal to the perimeter.

5. How do you use the circumference of a triangle in real life?

The circumference of a triangle can be used in real life for various purposes, such as calculating the distance around a triangular-shaped object, determining the length of fencing needed for a triangular-shaped garden, or finding the circumference of a circular-shaped object that can be inscribed in a triangle. It is also used in various mathematical and geometric applications.

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