jain2016 wrote:For the triangle shown above, where A, B and C are all points on a circle, and line segment AB has length 18, what is the area of triangle ABC?

(1) Angle ABC measures 30°.

(2) The circumference of the circle is 18.

Hi Experts ,

Please explain.

OAC

The key idea here is that an inscribed angle opposite the diameter of a circle is equal to 90 degrees. (The inscribed angle is half the central angle. If the central angle is a straight-line, or 180 degrees, the corresponding inscribed angle will be exactly half of that.) The other thing to note is that we can't simply assume that a line cutting through the middle of a circle is the diameter. We have to be told this.

S1: They're trying to tempt you into thinking that the triangle is a 30:60:90. But because we don't know if AB is the diameter, we don't know if angle ACB is 90 degrees. (Maybe it's 88 degrees or 92 degrees. Who knows?) So statement 1 alone is not sufficient.

S2: I'm assuming this should read that the circumference of the circle is 18Pi. If so, we now know that 18 is, in fact, the diameter of the circle, which means that the inscribed angle ACB is 90 degrees. But we don't know what kind of right triangle we're dealing with here. Could be 30:60:90. Could be something else. The type of triangle will dictate the lengths of the height and base. So this is also not sufficient on its own.

Together: we know that 18 is the diameter, and that angle ACB is 90 degrees. We know that angle ABC is 30, so angle CAB is 60. Because we know that the ratio of the sides of the 30:60:90 triangle is x: x * rt 3: 2x, we can find all the sides if we have the length of one side. If AB is 18, or 2x, then AC, or x must be 9, and CB must be 9 rt 3. Now we have the base and the height of the triangle, so we can find the area. Together the statements are sufficient. The answer is C