BTGmoderatorDC wrote:In a circle centered on point O with radius 8, diameter AC and point B on the circle, pictured above, what is the length of line segment (AB)?
(1) ∠AOB = 120°
(2) OB = BC
OA D
Source: Veritas Prep
The image is not attached.
Anyway, it's not needed.
So, B is any random point on the circle. Draw the figure as per the given information.
We have to get the value of AB.
Let's take each statement one by one.
(1) ∠AOB = 120°
Join OB. In ∆AOB, we have OB and AO as radii. Thus, AO = OB = 8/2 = 4. Thus, ∆AOB is an isosceles triangle. Thus, /_OAB = /_OBA. Given /_AOB = 120º, we have /_OAB = /_OBA = 30º. The length of AB can be calculated by splitting ∆AOB into two equal triangles of 90-60-30. We know that for a 90-60-30 triangle if the length of one side is known, we can get the value of all the sides. Sufficient.
(2) OB = BC
Join OB. We know that OB is radius; thus, OB = OC = 4. And, OC = OB = BC = 4; so, ∆OBC is an equilateral triangle. Thus, /_BOC = 60º, thereby /_AOB = 180 - 60 = 120º.
This is the same information that we got in Statement 1. Sufficient.
The correct answer:
D
Hope this helps!
-Jay
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