### Video Transcript

๐ด๐ต๐ถ is an equilateral triangle of side length 12 centimeters that is inscribed in a circle. Find the radius of the circle, giving the answer to two decimal places.

Letโs begin by sketching out a diagram. It doesnโt need to be perfectly to scale, but it should be roughly in proportion so we can check the suitability of any answers we get. Since the triangle is inscribed in a circle, that means the vertices must all lie on the circumference of the circle itself. We can construct the radii of the circle as shown. Now letโs add some angles. We know that the angles in an equilateral triangle are all 60 degrees. This means that the measure of the angle ๐๐ด๐ต must be half of this. Thatโs 30 degrees.

Similarly, the measure of the angle ๐๐ต๐ด must also be 30 degrees. Finally, since angles in a triangle sum to 180, we can calculate the measure of angle ๐ด๐๐ต by subtracting 30 and 30 from 180 to get 120 degrees. If we look solely at the non-right-angled triangle ๐ด๐๐ต, weโll see that we know the measure of three angles and the length of one of its sides. We can use the law of sines to calculate the missing lengths. We know itโs not the law of cosines since that requires at least two known sides.

Remember, the law of sines can be used in either form. Since weโre trying to find a missing length, we choose to use the first form. This form requires us to do less rearranging to solve any equations that we get. If we weโre trying to find a missing angle however, we would choose the second form. Letโs label the sides of our triangle. The side opposite the angle ๐ด is denoted by lowercase ๐. The side opposite angle ๐ is denoted by lowercase ๐. And the side opposite angle ๐ต denoted by lower case ๐.

Weโre trying to calculate the length of the radius of this circle. So weโre trying to find either ๐ or ๐. Letโs calculate the length of the side ๐. We know the measure of the angle at ๐ and the length of the side ๐, so weโll use these two parts of the formula: ๐ over sine ๐ด equals ๐ over sine ๐. Notice that weโve changed the letters to suit the triangle weโve drawn. The next logical step is to substitute any values we have from our triangle into the formula for the law of sines.

That gives us ๐ over sine 30 is equal to 12 over sine of 120. We can solve this equation by multiplying both sides by sine of 30. That gives us that ๐ is equal to 12 over sine of 120 multiplied by sine 30. Putting that into our calculator and we get a value of 6.9282. Correct to two decimal places, that gives us 6.93 centimeters as the radius of the circle. Itโs useful to know that we can check our answer with something called the extended law of sines.

This says that for a triangle inscribed in a circle, its ratio of its side length to the sine of the opposite angle is equal to two times the radius. Thatโs ๐ over sine ๐ด equals ๐ over sine ๐ต equals ๐ over sine ๐ถ, which is equal to two ๐. If we just choose two parts of this formula, letโs say ๐ over sine ๐ถ equals two ๐, we can quickly calculate the value of the radius. The measure of the angle at ๐ถ is 60 degrees. And the length of the side is 12.

So our formula will become 12 over sine 60 equals two ๐. We can solve this equation by dividing both sides by two. That gives us that the radius is six divided by sine 60, which is once again 6.9282 as we found out earlier. The radius of this circle then is 6.93 centimeters.