# Maximum Length

Two circles intersect at points C and D. Point A lies on Circle 1, point B lies Circle 2. Find the positions of A and B (on their given circle) such that segment AB is at its maximum possible length, and points A, C, and B are collinear. Circle 1                                      Circle 2

Answer

We first connect points A, B, and D to construct triangle ABD. What happens with the angles of the triangle as A and B change their position? Circle 1                                      Circle 2

Angle BAD is the same as CAD, and angle CAD is half the measure of arc (CD) (Interior Angle Theorem). Therefore as we move point A along its circle, the measure of angle BAD does not change.

Same argument can be made for angle ABD, and we can conclude that the size of the angles of triangle ABD does not change as A and B slide along the two circumferences. Circle 1                                      Circle 2

Any two versions of ABD are similar triangles, and the version with the longest possible length of AB has the longest possible lengths for AD and BD too.

When is AD the longest? When A and D are the end points of a diameter of Circle 1.

When is BD the longest? When B and D are the end points of a diameter of Circle 2. What is the measure of the angle determined by AB and CD? Both angle ACD and angle BCD are inscribed in a semicircle, so they measure 90°.

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