Question 1180131
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The triangle is isosceles.  The altitude to the 10-inch side divides the triangle into two right triangles each with hypotenuse 13 and one leg 10/2=5.  By the Pythagorean Theorem, the other leg (the altitude of the triangle) is 12.<br>
The center of the circumscribed circle is on that altitude.<br>
Let x be the radius of the circumscribed circle; the distance from the center of the circle to each vertex of the triangle is then x.<br>
The distance from the center of the circle to the foot of the altitude is then 12-x.<br>
The radius to one of the base vertices, along with portions of the base and altitude of the triangle, form a right triangle with legs 5 and 12-x and hypotenuse x.<br>
{{{drawing(400,400,-6,6,-2,14,
line(-5,0,5,0),line(0,0,0,12),line(-5,0,0,12),line(5,0,0,12),
line(-5,0,0,119/24),
locate(-2.5,-.5,5),locate(2.5,-.5,5),locate(-3.5,6,13),locate(-.5,8,x),locate(-2.5,4,x),locate(.5,3,"12-x")
)}}}
Use the Pythagorean Theorem to solve for the radius x; then double that to find the diameter.<br>