Math CentralQuandaries & Queries


Question from eujane, a student:

in triangle ABC in which AB=12,BC=18,&Ac=25, a semicircle is drawn so that its diameter lies on AC and so that it is tangent to AB and BC.if O is the center of the circle, find the measure of OA.9

Call D and E the points where the tangents BC and BA touch the circle. Then OD = OE because they are radii, and BD = BE because they are tangents from the point B. That tells us that the quadrilateral OEBD is symmetric about the line OB; in other words, OB bisects angle CBA. (Draw an accurate figure!) Now use the theorem that tells us that the bisector of angle B divides the segment CA in the same ratio as the sides BC and BA, namely


Combine this with the given information that CO + OA = CA = 25.


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