Results 1 to 3 of 3

Math Help - triangle inscribed in the circle

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    173

    triangle inscribed in the circle

    triangle ABC is inscribed in the circle and AC = AB. The measure of angle BAC is 42 degrees and segment ED is tangent to the circle at point C. What is the measure of angle ACD?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by sri340 View Post
    triangle ABC is inscribed in the circle and AC = AB. The measure of angle BAC is 42 degrees and segment ED is tangent to the circle at point C. What is the measure of angle ACD?
    If you draw a circle, locate the centre, then the inscribed triangle is isosceles.
    Join the centre to all 3 triangle vertices.
    The angle at A is now split into 2 equal 21 degree angles.
    If we label the centre F, then angle FAB = angle FAC = 21 degrees.
    The 3 triangles within the inscribed triangle ABC are all isosceles also.
    Hence, angle FCA = angle FBA = 21 degrees.

    The tangent is perpendicular to the line CF.
    Therefore the sum of angles FCA and ACD is 90 degrees.
    Hence angle ACD is 90-21 = 69 degrees.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,860
    Thanks
    742
    Hello, sri340!

    Great solution, Archie!

    Here's another approach . . .


    Triangle ABC is inscribed in the circle and AC = AB.

    The measure of \angle BAC is 42
    . . and segment ED is tangent to the circle at point C.

    What is the measure of \angle ACD ?
    Code:
                    A
                  * o *
              *    / \    *  138
            *     /42\     *
           *     /     \     *
                /       \         o D
          *    /         \    *  /
          *   /           \   * /
          *  /             \  */
            / 69           \ /
         B o- - - - - - - - -o C
            *               *
              *           */
                  * * *   /
                         /
                        o E

    Since \Delta ABC is isosceles, \angle B = \angle C = 69^o

    Then \text{arc}(AC) = 138^o
    . .
    An inscribed angle is measured by one-half its intercepted arc.

    Therefore: . \angle ACD \,=\,69^o
    . .
    The angle between a common tangent and secant
    . . . . is measured by one-half its intercepted arc.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Triangle inscribed in circle
    Posted in the Trigonometry Forum
    Replies: 8
    Last Post: October 11th 2010, 04:21 AM
  2. triangle inscribed in a circle...
    Posted in the Geometry Forum
    Replies: 1
    Last Post: February 26th 2010, 01:48 PM
  3. Triangle inscribed in one Circle
    Posted in the Geometry Forum
    Replies: 1
    Last Post: February 7th 2010, 08:43 PM
  4. Replies: 2
    Last Post: February 6th 2010, 09:31 AM
  5. Replies: 27
    Last Post: April 27th 2008, 11:36 AM

Search Tags


/mathhelpforum @mathhelpforum