Question

In the accompanying diagram, right triangle ABC is inscribe in the circle 0 and AOB is a diameter. If

AC = 6 and BC = 2√√3, find the exact length of the radius of the circle.
Answer in simplest form.

Answer 1

In order to find the radius of the circle, we need to use the fact that the circumcenter of a right triangle is the midpoint of the hypotenuse. Since AOB is a diameter, it follows that O is the midpoint of AB. Let's call the radius of the circle r.

We can use the Pythagorean theorem to find AB:

AB^2 = AC^2 + BC^2

AB^2 = 6^2 + (2√3)^2

AB^2 = 36 + 12

AB^2 = 48

AB = 4√3

Since O is the midpoint of AB, we have AO = BO = AB/2 = 2√3.

Now we can use the formula for the area of a triangle to find r:

Area(ABC) = (1/2) * AC * BC

Area(ABC) = (1/2) * 6 * 2√3

Area(ABC) = 6√3

r = Area(ABC) / (1/2 * AB)

r = (6√3) / (1/2 * 4√3)

r = (6√3) / (2√3)

r = **3**

Therefore, the exact length of the radius of the circle is **3**.