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Measure that angle (Posted on 2002-06-18) Difficulty: 5 of 5
Given that:
  • ABC is an isosceles triangle in which
        AB = AC
  • The lengths of the following segmets are equal:
        AD
        DE
        EC
        BC
    Find the measure of angle A.
  • See The Solution Submitted by vohonam    
    Rating: 4.2000 (20 votes)

    Comments: ( Back to comment list | You must be logged in to post comments.)
    Solution Puzzle Solution With Explanation Comment 59 of 59 |
    (In reply to Answer by K Sengupta)

    By conditions of the problem, we obtain:

    A + 2B = 2*pi rad, where A = < BAC and:
    < ACB = < ABC = B .......(I)

    Also, since Triangle AED is isosceles, it follows that:
    < AED = A, while < BDE = < AED + < ADE = 2A

    Again, < CED = 2*pi - A = 2B in terms of (i)

    We now draw BP which bisects < BCE and meets DB at the point Q. We join PQ.

    Thus, triangles QCP and QCB are congruent, and accordingly:

    < QPC = < QBC = B

    But, < DPC = < DPQ + < CPQ
    Or, 2B = < DPQ + B, giving:
    < DPQ = B

    Thus, it follows that the triangles DPQ and CPQ are congruent, giving:

    < QDP = < QCP

    But, < QDP = < QCP = 2A, and:
    < QCP = = B/2, so that:
    2A = B/2
    Or, B = 4A

    Substituting  this into equation (i), we obtain:
    9A = 2*pi rad, giving A = 2*pi/9

    Consequently, the required measure of the angle A is 2*pi/9 radians. 

    Edited on August 21, 2007, 11:15 am
      Posted by K Sengupta on 2007-05-31 05:19:37

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