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Question 633696: If one side of a triangle is 12 inches and the opposite angle is 30 degrees, find the diameter of the circumscribed circle.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Here is the triangle, with the circumscribed circle O, and a few useful rays to shine light on the problem:
 The triangle is ABC, with a  degree angle at C, and AB=12 inches..
The central angle, AOB, has a measure of  because, in a circle,
the measure of a central angle (AOB) is twice the measure of the inscribed angle (ACB) that intercepts the same arc.
OA and OB are radii, congruent with each other, so we figure that AOB is very symmetrical,
at least isosceles, with congruent base angles at A and B.
Ray OD is the bisector of angle AOB and splits angle AOB into two congruent  angles (AOD and DOB).
It also splits triangle AOB into two triangles (OAD and OBD).
Those triangles, having congruent sides forming congruent angles, are congruent.
The angles at D (ADO and BDO) are also corresponding parts of congruent triangles, which makes them congruent.
Since AB is a straigh line, angles ADO and BDO are right angles.
That means that OAD and OBD are right 30-60-90 triangles.
That makes AOB more than isosceles; since it has three  angles, it's equilateral.
That means that radii OA and OB measure 12 inches, just like AB.
The diameter of the circle is twice radius OA, or  inches in length.
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