An equilateral triangle is inscribed in a circle with r = 10. Find the perimeter of the triangle.

Question

Find the perimeter of the triangle. 

Answer 1

Hi Geny,

 

In order to calculate the perimeter of the equilateral triangle inscribed in the circle, we'll first need to determine the length of one of its sides.  Let's see how we can do this:

    1.  If we were to draw three radii to each vertex of the triangle, we discover that they form three isosceles triangles in which each triangle has a vertex angle of 120 degrees (corresponding to the measure of each the three equal arcs cut off by each side of the equilateral triangle).

    2.  Each of the isosceles triangles will also each have congruent base angles (measure of 30 degrees each).

    3.  If we were to draw a radius perpendicular to the base of one of these isosceles triangles, we know that it will bisect the base (a side of the inscribed equilateral triangle).

    4.  We can now calculate the length of half of a  side of the equilateral base using the 30-60-90 right triangle side ratio:  side opposite 60 degree angle is half the length of the hypotenuse times radical 3, which in our case is half the length of the radius times radical 3 (1/2 x 10 radical 3 = 5 radical 3).

    5.  The length of a full side of the equilateral triangle will be twice the length of our calculated half side (2 x 5 radical 3 = 10 radical 3).

 

Now we can find our answer (perimeter of the inscribed equilateral triangle) by multiplying the length of our calculated side by 3:  3 x 10 radical 3 = 30 radical 3.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.

Answer 2

Here's another solution to the problem similar to Jordan's.

Draw a diameter from a vertex of the triangle.

The diameter subtends a right angle.

Draw the third, smaller side of the right triangle.

The diameter is the hypotenuse and the longer leg is one side of the triangle.

The diameter bisects the 60 degree angle.

The hypotenuse is 20, the smaller side is opposite the 30 degree angle so it is 10.

Call the longer side, one of the sides of the triangle, x.

10^2+x^2=20^2

100+x^2=400

x^2=300

x=√300

x=10√3

3x=30√3, the perimeter