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Question

What is the sum of all possible values of t between 0 and 360 such that the triangle in the coordinate plane whose vertices are ( 40^, 40^), ( 60^, 60^), and ( t^, t^) is isosceles?

Accepted Answer

Let A = ( 40^, 40^), B = ( 60^, 60^), and C = ( t^, t^). We apply casework to the legs of isosceles ABC: 1. AB=AC Note that A must be the midpoint of BC. It follows that C = ( 20^, 20^), so t=20. 2. BA=BC Note that B must be the midpoint of AC. It follows that C = ( 80^, 80^), so t=80. 3. CA=CB Note that C must be the midpoint of AB. It follows that C = ( 50^, 50^) or C = ( 230^, 230^), so t=50 or t=230. Together, the sum of all such possible values of t is 20+80+50+230=: 380. Remark The following diagram shows all possible locations of C: FIGURE