Bunuel wrote:
In the circle above, chord AC = 85. What is the area of the circle?
(1) Angle ABC = 90°
(2) AB = 51 and BC = 68
Kudos for a correct solution. MAGOOSH OFFICIAL SOLUTION:If we know chord AB were a diameter, then we would divide by 2 to find the radius. Once we know the radius, we could find the area. Furthermore, we would know that AB were a diameter if the angle at B were a right angle, because an
inscribed angle of 90° always intersects a semicircular arc.
Statement #1 simply tells us that the angle at B is 90°, so we are done! This statement, alone and by itself, is sufficient.
Statement #2 gives us the other two sides of triangle ABC, we now we know we have a {51, 68, 85} triangle. If these three sides satisfy the
Pythagorean Theorem, then the triangle would be a right triangle, and the angle at B would be 90°. Well, it would be HUGE mistake to square these numbers as is and try to do anything with the results! Instead, can we factor out a common factor? Well, 51 = 3*17. Also, notice that 85 = 5*17. This makes us suspicious, and it turns out, 68 = 4*17. This is simply the good old 3-4-5 triangle, with everything multiplied by 17. Of course, this is a right triangle, so the angle at B is 90°. This statement, alone and by itself, is sufficient.
Answer = (D)