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B Couple geometry/trigonometry questions

  1. May 15, 2016 #1
    I am reading Gelfand's Trigonometry. In one of the questions he asks: "We know from geometry that a circle may be drawn through the three vertices of any triangle. Find the radius of such a circle if the sides of the triangle are 6,8, and 10."

    My first question is, I know that if the diameter of a circle is the hypotenuse of a triangle then that triangle is a right triangle. Does this imply that the hypotenuse of any right triangle inscribed within a circle must be the diameter?

    If this is not the case then I'm at a loss on how to solve this problem.

    Second question, I have searched around but cannot find solutions to this book is there a place to find the solutions.
    Last edited: May 15, 2016
  2. jcsd
  3. May 15, 2016 #2


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    Yes, it must be the diameter.
  4. May 15, 2016 #3


    Staff: Mentor

    Unfortunately I cannot answer your last question. The answer to the first, however, is yes. Imagine you have the hypotenuse of a right triangle in a circle and it is not the diameter. Then for the third point to be on the circle you get either a longer side which cannot be true or an angle which cannot be right which cannot be true either.
  5. May 15, 2016 #4
    That makes sense, so the answer to Gelfand question quoted above would just be 5. Seems like the question is too easy which is why I asked my question in the first place. Well thank you for your response.
  6. May 15, 2016 #5


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    I think the author means a circle inscribed in a triangle, instead of the opposite. He said "sides", not "corners".
    If assuming the circle inscribed in a triangle is correct, then will this animation be helpful?
  7. May 15, 2016 #6
    Actually my mistake in quoting the question. It actually states"... may be drawn the the three vertices of any triangle". Sorry about that don't know how I mistyped that. Fixed my original question.
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