Given the circle below, with center O, diameter AOB,

Hi All,

We're told that in the circle with center O, AOB is the diameter, the radius of the circle is 5 and triangle ABC is inscribed. We're asked for the the length of AC. This question is based on some specific Geometry patterns - and if you recognize them, then you can answer this question without doing much math at all.

To start, any triangle in a circle that has all 3 vertices ON the circumference AND has one side that is the diameter of the circle is a RIGHT TRIANGLE. Thus, Angle C is a 90-degree angle.

Next, since the radius of the circle is 5, the diameter is 10. We now know two of the sides of the right triangle (8 and 10 - which is the hypotenuse). You can use the Pythagorean Theorem (A^2 + B^2 = C^2) to find the missing side OR you might recognize that we're dealing with a 3/4/5 right triangle that has been 'doubled' (into a 6/8/10 right triangle). Thus, the missing side is 6.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

VJesus12 wrote:Given the circle below, with center O, diameter AOB, a radius of 5, and the inscribed triangle ABC, what is the length of AC?



A. 4.5
B. 5
C. 6
D. 6.5
E. 7

[spoiler]OA=C[/spoiler]

Source: Manhattan GMAT

Since the angle ACB is an inscribed angle subtended by a diameter, angle ACB is a right angle. Since we have a right triangle with a hypotenuse of 10, we see that we have a 6-8-10 right triangle. Thus, AC = 6.

Answer: C