1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show 2 of 3 points on a circle are the diameter

  1. Nov 22, 2012 #1

    trollcast

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Points A B and C lie on the circumference of a circle where
    $$A =(-3,2)\\B=(-1,6)\\C=(7,2)$$

    Show that AC is the diameter of the circle.

    2. Relevant equations

    3. The attempt at a solution

    Would it be sufficient to show that the angle ABC is a right angle and therefore by the inscribed angle theorem that any angle subtended from a diameter is a right angle?
     
  2. jcsd
  3. Nov 22, 2012 #2
    Consider a point O, such that O is the midpoint of AC. If AC is the diameter of the circle, then any point on the circumference of the circle should be equidistant from point O.

    I think you can finish from there.
     
  4. Nov 22, 2012 #3

    trollcast

    User Avatar
    Gold Member

    Thanks I used a similar solution by finding a mid point of AC and then showing that OB was the same length as OC?

    So is the diameter not the only 2 points you can subtend right angle triangle from then? (I can't picture any other cases)
     
  5. Nov 22, 2012 #4
    That's right.

    Naw. I just find it simpler to use less theorems.
     
  6. Nov 22, 2012 #5

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Not exactly. You are trying to show a certain length is a diameter, so you would need a theorem that says the only chords which subtend a right angle on the circumference are diameters. In fact, I believe the inscribed angle theorem says that, but that's not how you quoted it.
     
  7. Nov 22, 2012 #6

    symbolipoint

    User Avatar
    Homework Helper
    Education Advisor
    Gold Member

    Coordinate geometry for finding equation for the circle which contains those three points could be easier to handle. The resulting and simplified equation will show the size of the radius; then, you merely use distance formula to show that the intended given points (giving length, AC) are twice the size of radius.
     
  8. Nov 22, 2012 #7
    ...was my answer wrong? :confused:
     
  9. Nov 22, 2012 #8

    symbolipoint

    User Avatar
    Homework Helper
    Education Advisor
    Gold Member

    No. In fact, maybe a better method is to understand that if two of the points are the endpoints of a diameter, then the three points given, being all on the circle, form a right-triangle. The longer side would be the hypotenuse, and therefore the circle's diameter.

    My earlier posted method of coordinate geometry to try to make a system of three equations would not be very neat to solve.
     
  10. Nov 22, 2012 #9

    symbolipoint

    User Avatar
    Homework Helper
    Education Advisor
    Gold Member

    Your idea now seems a correct way. You can also find that the slopes of two of the sides are opposite-reciprocals of each other (meaning, intersect at right-angle). You can find the midpoint of the longest side and use this to fill-in the standard equation of a circle and show that the three points satisfy the equation.

    EDIT: Just having solved this most of the way through, what I suggest is very good; but not necessary to check the slopes of any segments. You can if you wish.
    IF longest side is diameter, then its midpoint is the center of the circle. Longest side length indicates the diameter from which you get the size of radius. You use the found center point of this longest side to fill-in the standard form of equation for a circle. NOW, you can check to see if each point satisfies the equation.
     
    Last edited: Nov 22, 2012
  11. Nov 22, 2012 #10

    Mentallic

    User Avatar
    Homework Helper

    I'm not sure why everyone has been leading the OP away from his solution. Showing angle ABC to be a right angle is an ingenious solution, considering it's very simple to do and it's short and sweet.
     
  12. Nov 22, 2012 #11

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    For my part, because the OP ends thus: "Would it be sufficient to show that the angle ABC is a right angle and therefore by the inscribed angle theorem that any angle subtended from a diameter is a right angle?"
    The logic of that last part is backwards from what's needed here, as in "A implies B" versus "B implies A". It just has to be changed to "any chord that subtends a right angle at the circumference is a diameter" (that is, assuming that the 'inscribed angle theorem' says that).
     
  13. Nov 22, 2012 #12

    Mentallic

    User Avatar
    Homework Helper

    I felt like the OP was just giving us a summary of his reasoning, merely asking a question to see if he was on the right track, but I agree with you that it's very easy to make that logical mistake that you pointed out and it's good to see that you're explicitly critiquing it.

    Oh, and it's not your part I was particularly concerned with :wink:

    But I realize now it was a bad choice of words to say that "everyone" has been leading the OP away...
     
  14. Nov 23, 2012 #13

    trollcast

    User Avatar
    Gold Member

    Thanks,

    I think I maybe got the circle theorem wrong in my OP

    http://en.wikipedia.org/wiki/Thales'_theorem

    Is that basically what I was talking about showing ABC to be right angled would be sufficient to show the AC is the diameter?
     
  15. Nov 23, 2012 #14

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    The result you need is the one described as the converse at that link.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Show 2 of 3 points on a circle are the diameter
  1. Diameter of a Circle (Replies: 9)

Loading...