In the figure above (not drawn to scale), the triangle

*ABC* is inscribed in the circle with the center

*O* and the diameter

*AB*.

*AC* is 4 inches shorter than

*AB*. The ratio of the angle

*COB*’s measure to the measure of angle

*OBC* is 2:1. What is the length of

*OB*?

A. 2√2

B. 2√3

C. 4

D. 4 + 2√2

E. 4 + 2√3

(D) Let us use the fact that the ratio of the angle COB’s measure to the measure of the angle OBC is 2 to 1. Denote the measure of the angle OBC as

*x* degrees. Then, the measure of the angle COB is 2

*x* degrees. OB = OC, as the both are radii. Therefore the triangle BOC is isosceles and angle OCB = angle OBC =

*x* degrees. Since the sum of all the angle measures in any triangle is 180 degrees, we can write the equation:

*x* +

*x* + 2

*x* = 180

4

*x* = 180

*x* = 45

So, the measure of the angle BOC is 2

*x* = 90 degrees. The angles AOC and COB are contiguous so angle AOC = 180° – 90° = 90°. Also AO = OC as radii. Therefore, triangle AOC is an isosceles right triangle (90°-45°-45°). The sides of such triangle have 1 : 1 : √2 ratio.

The question statement defines AC = AB – 4. Since AB is a diameter and AO is a radius AC = 2AO – 4. When we plug it in AO : AC = 1 : √2 we get AO : (2AO – 4) = 1 : √2 .

√2AO = 2AO – 4

4 = 2AO – √2AO

4 = (2 – √2)AO

AO = 4/(2 – √2)

AO = 4(2 + √2) / ((2 – √2)(2 + √2))

AO = (8 + 4√2) / 2

AO = 4 + 2√2

The correct answer is D.

If you don't remember the property of a 90°-45°-45° triangle, there is an alternative ending of the solution. Let us use the fact that AC is 4 inches shorter than AB. Denote radii by

*y* inches. So, we can deduce three things:

1. AO = OB = OC =

*y*2. AB = 2

*y*3. AC = 2

*y* – 4

Then we use the Pythagorean Theorem for triangle AOC and get the following equation:

(2

*y* – 4)² =

*y*² +

*y*²

4

*y*² – 16

*y* + 16 = 2

*y*²

2

*y*² – 16

*y* + 16 = 0

*y*² – 8

*y* + 8 = 0

*y* = 4 + 2√2 and

*y* = 4 – 2√2 are two solutions to this equation. But, the length of AC (2

*y* – 4) must be a positive number, so

*y* = 4 + 2√2 is the only option.

The correct answer is D.

----------

since the triangle AOC is right isoceles, can we simply use the Pythagorean Theorem 1:1:√2?

1:√2 = AO:4, AO = 4√2 /2 = OB