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The moment the three numbers appear, a pythagorean triplet flashes in the mind. A small calculation confirms the inkling.
So now, we are talking about an angle in a semicircle. Basic geometry tells us that the angle in a semicircle is always a right angled triangle. Now make both ends meet. Since the right angled triangle is circumscribed the next obvious point is that the hypotenuse (which is also the longest side = 41) of this triangle must be the diameter. Hence the 20.5 is the right answer without a second thought. Hope this helps.
What is the measure of the radius of the circle that circumscribes a triangle whose sides measure 9, 40 and 41? (A) 6 (B) 4 (C) 24.5 (D) 20.5 (E) 12.5
First of all we can notice that a triangle whose sides measure 9, 40 and 41 is a right triangle because 9^2 + 40^2 = 41^2.
Next, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle (the reverse is also true: if the diameter of a circle is also the triangle’s side, then that triangle is a right triangle).
Thus the diameter of the circle is the hypotenuse of the triangle --> diameter = hypotenuse = 41 --> radius = 41/2 = 20.5.
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