Perimeter of a triangle inscribed in a circle

[Fedcer]

Homework Statement

Find the Perimeter of an equilateral triangle inscribed in a circle knowing the radius r.

Homework Equations

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The Attempt at a Solution

Browsing the web I found that the intersection of the three perpendicular bisectors of a traingle is the center of it's circumscribed circle. Knowing this I proceeded as following: [img=http://img214.imageshack.us/img214/2971/problem1bgc6.th.jpg]
I think that what I did is correct. However, since I should've been able to solve it with what I already knew I thought that maybe there is another easier solution.

Thanks.

 

[jing]

Triangle AOB is isoceles. Just drawing the perpendicular from O to AB get you to the same position as the result you quote ( in fact it demonstrates it)

 

[Architeuthis Dux]

I would like to commend your attempt at solving this problem. Your use of the concept of perpendicular bisectors and the center of a circumscribed circle is a valid approach. However, I would like to suggest an alternative method that may be simpler and more intuitive.

First, we know that an equilateral triangle has all three sides equal in length. In this case, the triangle is inscribed in a circle, so each side will be a chord of the circle. We can use the formula for the length of a chord in a circle, which is c = 2r sin(θ/2), where c is the chord length, r is the radius of the circle, and θ is the central angle of the chord.

Since we know that the central angle of an equilateral triangle is 120 degrees, we can plug this value into the formula and get c = 2r sin(120/2) = 2r sin(60) = 2r √3/2 = r√3. This gives us the length of one side of the triangle.

Since all three sides are equal, the perimeter of the triangle will be 3 times this length, which is 3r√3.

In summary, the perimeter of an equilateral triangle inscribed in a circle with radius r is 3r√3. This approach may be more straightforward and efficient in solving the problem.