Results 1 to 4 of 4

Math Help - Circle

  1. #1
    Junior Member
    Joined
    May 2009
    Posts
    68

    Circle

    Point P is 20 unites from the centre of a circle of radius 29. The number of different chords of the circle through P with inteher lengths is
    a] 16
    b] 17
    c] 32
    d] 33
    e] 34

    help and explaination pls?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by foreverbrokenpromises View Post
    Point P is 20 unites from the centre of a circle of radius 29. The number of different chords of the circle through P with inteher lengths is
    a] 16
    b] 17
    c] 32
    d] 33
    e] 34

    help and explaination pls?
    I think the answer is (c). First draw a picture of a circle and mark the points P and O(centre). Now convince yourself that the shortest length chord passing through P is the one thats perpendicular to OP.Now compute the length of the chord through Pythogorus theorem. The chord length will be 42. SInce the largest length chord is the diameter, length of the largest length chord through P is 58. Thus the chords range from 42 to 58. Now if you carefully count you will see that the number of chords is two times the number of chords on one side and the answer is 32
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jan 2009
    Posts
    591
    Quote Originally Posted by foreverbrokenpromises View Post
    Point P is 20 unites from the centre of a circle of radius 29. The number of different chords of the circle through P with inteher lengths is
    a] 16
    b] 17
    c] 32
    d] 33
    e] 34

    help and explaination pls?
    Suppose that the circle is centered at the origin (0,0), and that the point P is located at (20,0).

    The first chord passing through the point P will be from (29,0) to (-29,0) with a length of 58.
    Another chord will be from (-29,0) to (29,0) or in the opposite direction, but will still have the same length of 58.
    I suppose that only makes a chord count (with integer length ) of 1 since they both are equal to 58.

    That is the longest possible chord.


    The shortest possible chord will occur when a point on the circle occurs at (20,y) or (20,-y).
    [That creates a right triangle]
    The radius (or hypotenuse) is 29,
    the base is 20,
    the height is y

     y = \sqrt{29^2 - 20^2} = 21

    Thus the chord will be from (20,21) to (20,-21) for an integer length of 42.

    That is the shortest chord possible.


    The chord length will vary (continously) from 42 to 58. You will have a chords of various lengths between 42 to 58 inclusive.


    Simply count the number of integers from 42 to 58 (including 42 and 58).

    That should be your answer

    Spoiler:
    17
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Here's a picture to illustrate the answer. The longest chord (length 58) is the diameter AB. The shortest is the perpendicular chord CD (length 42). These only occur once. Every intermediate length occurs twice, once as X moves along the arc from B to C, and once as X moves along the arc from C to A. The total number is that given by Isomorphism.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Circle, tangent line, and a point not on the circle
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: March 31st 2011, 02:40 PM
  2. Replies: 6
    Last Post: July 8th 2010, 06:39 PM
  3. Replies: 7
    Last Post: March 15th 2010, 05:10 PM
  4. Replies: 2
    Last Post: February 6th 2010, 09:31 AM
  5. Replies: 0
    Last Post: October 12th 2008, 05:31 PM

Search Tags


/mathhelpforum @mathhelpforum