Triangle Circumscribed by a Circle (Geometry Problem)

In summary, the conversation revolves around finding the length L of an equilateral triangle that touches a circle of radius R. The approach involves drawing a 60 degree angle and bisecting it to form a right triangle, and then creating a similar triangle by drawing a perpendicular line. There is a discussion about the necessary information needed to write L in terms of R, and someone provides a hint by pointing out a triangle with a 120 degree angle and another triangle with a 30-60-90 right triangle. The conversation ends with a realization that drawing another radius line would have helped in solving the problem.
  • #1
Saladsamurai
3,020
7

Homework Statement



Alright... so I thought of this problem on the train ride home. I am sure plenty thought of it before me...but now it is my turn.

Say want to know what the length L an equilateral triangle must have such that all three points touch the circle of radius R that it is contained by.

I have drawn the problem. i have bisected one of the 60 degree angles to form a right triangle.

I then drew a line perpendicular to the bisecting line to create a similar triangle.

I do not know if this is the way you would have approached this, but it seemed pretty logical to me.

I feel like I have almost all (if not all) the necessary info to write L in terms of R.

Can someone chime in with a hint here?

Picture1-25.png
--->
Picture4-4.png


I feel like I need something more about either the short leg of the inner triangle or its hypoteneuse. then I can relate
both triangles to the angle.
 
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  • #2
You just want the relation between L and R? I see in your picture a triangle with a 120 degree angle at the center between two legs of length R and with an opposite side of length L. It's just trig, saladsamurai, I know you know that. It must have been a rough train ride home.
 
  • #3
I presume this is what you meant Dick:

Picture5-3.png


I can honestly say that I may have never seen that if you did not point it out :/

I have never taken a geometry class in my life, so these things do not pop out at me sometimes... arggggghhhh

Thank Dick!
Casey
 
  • #4
You didn't even need to see it. I also see a 30-60-90 right triangle with a hypotenuse of R and a leg of length L/2. I'm still blaming the train ride home.
 
  • #5
Now you're just showing off after I said I suck at Geometry! I STILL do not see that one!

edit: Okay NOW I see it! But I never would have since I still would have needed to have drawn that same line that allowed me
to see the 120 degree triangle.
 
  • #6
Yeah. You would have gotten it just fine if you'd drawn another radius line. Remember that when you're stuck in the future.
 

1. What is a Triangle Circumscribed by a Circle?

A Triangle Circumscribed by a Circle is a geometric problem where a triangle is drawn inside a circle in such a way that all three vertices of the triangle lie on the circumference of the circle.

2. How is this problem solved?

This problem can be solved using a few different methods, such as using the Pythagorean theorem, the Law of Sines, or the Law of Cosines. It ultimately depends on the given information and what is being solved for.

3. What is the relationship between the triangle and the circle?

The triangle is said to be "circumscribed" by the circle because all three vertices of the triangle lie on the circumference of the circle. In other words, the circle is drawn in a way that it passes through all three vertices of the triangle.

4. Can a triangle be circumscribed by any circle?

Yes, a triangle can be circumscribed by any circle as long as the circle's center is located at the midpoint of one side of the triangle. This is known as the circumcenter of the triangle.

5. What is the significance of solving this problem?

Solving this problem can help in understanding the relationship between circles and triangles, as well as improve problem-solving skills and logical thinking. It is also often used in real-world applications, such as in engineering and architecture.

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