Circumscribed circles
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The circumscribed circle is the circle passing through the three vertices of a triangle ABC. Assume the following results from geometry. The perpendicular bisectors of the sides of a triangle meet in a point O that is the center of the circumscribed circle.
a) According to a theorem from geometry, the measure of the angle AOC is twice that of angle B. What theorem is this? (State the theorem in complete sentences)
b) Draw a perpendicular from the center O to AC, meeting AC at T. Denoting the length of the side opposite to the angle B by b, explain why AT=TC-b/2.
c) Explain why triangle ATO is congruent to triangle CTO.
d) Use the results in parts (a) and (c) to show that angle COT = angle B.
e) Us the result in part (d) to show that the radius R of the circumscribed circle for triangle ABC is equal to b/(2 sin B). The use the law of sines to conclude from this that
R = ___a_____ = _____b______ = _____c_____
2 sin A 2 sin B 2 sin C
f) Let A denote the area of the triangle ABC. Show the radius R of the circumscribed circle is given by
R = __abc___
4A
hint: use the result from part (e)
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Solution Summary
This uses a circle circumscribed through the vertices of a triangle, and answers questions regarding congruency and angle measures.
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a. The theorem is: the angle at the center is twice of the angle of circumference for a given chord in a cicle.
b. In your problem, the side opposite to the angle is . So . Since is ...
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