Question

An equilateral triangle with sides of length 6 is inscribed in a circle. What is the diameter of the Circle?
A:5.2
B:6
C:6.9
D:7.5

Answer 1

Answer:

The diameter is 6.92 unit.

Step-by-step explanation:

Given : An equilateral triangle with sides of length 6 is inscribed in a circle.

To find : What is the diameter of the Circle?

Solution :

First we picture the situation.

Refer the attached figure below.

We that the angles of equilateral triangle is 60 degrees.

Then, we divide the triangle into two parts to solve the radius.

As,

∠ACD is 60°

$\angle OCD=\frac{\angle ACD}{2}$

$\angle OCD=\frac{60}{2}=30^\circ$

In triangle OCD,

$\cos 30^\circ= \frac{CD}{OC}$

$\cos 30^\circ= \frac{CD}{OC}$

$\frac{\sqrt{3}}{2}=\frac{3}{r}$

$r=\frac{3\times 2}{\sqrt{3}}$

$r=2\sqrt{3}$

$r=3.4641$

Diameter = 2r=2(3.4641)=6.92 unit.

Therefore, The diameter is 6.92 unit.

Answer 2

The diameter of the circle is approximately 6.9 (Option C) and this is arrived at using the principles of an Equilateral Triangle and the Radius of a  Circle.

What is an Equilateral Triangle?

An equilateral triangle is a triangle whose sides are all equal. Because the triangle is in the circle, we can obtain the radius of the circle by bilaterally dividing the triangle using perpendicular bisectors. That way the equilateral triangle has three equal parts.

Recall that the sum of angles in a triangle is 180°.

This means, as shown in the attached image, that ∠ ACD is 60°

If that is the case, then ∠OCD = ∠ACD/2 = 60°/2 = 30°

To find the Radius, we can use the rule of Cosine.

Cos 30° = CD/OC

$\sqrt{3}$/2 = 3/r

r therefore = (3 x2)/$\sqrt{3}$

r = $\sqrt[2]{\sqrt{3} }$

Therefore, Radius (r) = 3.4641

Because diameter is equal to  2 x r,
diameter therefore is 3.4641 x 2 = 6.93;

which is approximately 6.9.

Learn more about Equilateral Triangels at:

https://brainly.com/question/17264112