Saturday, October 31, 2009

Problem 382. Concave Quadrilateral, Angle Bisectors, Angles

Proposed Problem
Click the figure below to see the complete problem 382 about Concave Quadrilateral, Angle Bisectors, Angles.

 Problem 382. Concave Quadrilateral, Angle Bisectors, Angles.
See more:
Complete Geometry Problem 382
Level: High School, SAT Prep

3 comments:

  1. Extend CE & CD to meet AB in F & G resply.
    angle GFC= α + C/2 and angle x = angle GFC + A/2
    Hence, x = α + C/2 + A/2 or 2(x- α) = A + C
    Likewise, angle AGC = α + C and β = ang. AGC + A or β = α + C + A Or β = α + 2(x- α) = 2x – α or x = (α+ β)/2
    Ajit

    ReplyDelete
  2. Let alpha=a, beta=b.
    Let measure angle EAD=z and measure angle ECD=w.
    ABCD is a concave quadrilateral,
    hence, b = a + 2z + 2w...(1)
    ABCE is a concave quadrilateral,
    hence, x = z+ w + a...(2)
    (a + b)/2=(a+a+2z+2w)/2...(by (1))
    =2(a+z+w+)/2
    =a+z+w
    =x...(by (2))

    ReplyDelete
  3. [For easy typing, I use a for alpha & b for beta]
    Join AC

    In triangle DAC
    b=180-<DAC-<DCA

    In triangle EAC
    x=180-<DAC-<DCA-<EAD-<ECD
    x=b-<EAD-<ECD
    <EAD-<ECD=b-x

    In triangle BAC
    a=180-<DAC-<DCA-<EAD-<ECD-<BAE-<BCE
    a=b-(b-x)-(b-x)
    a=2x-b
    a+b=2x
    x=(a+b)/2

    ReplyDelete