One of the sides of a triangle that inscribe in a circle is 6. Is the triangle a right triangle?
(1) Two of the sides of the triangle have the same length. Know nothing about the circle. Not sufficient.
(2) The circumference of the circle is 6 pi --> 2\pi{r}=6\pi --> r=3 --> diameter=6. So, one of the sides of the triangle is the diameter of the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s side, then that triangle is a right triangle. Sufficient.
Answer: B.
(1) Two of the sides of the triangle have the same length. Know nothing about the circle. Not sufficient.
(2) The circumference of the circle is 6 pi --> 2\pi{r}=6\pi --> r=3 --> diameter=6. So, one of the sides of the triangle is the diameter of the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s side, then that triangle is a right triangle. Sufficient.
Answer: B.