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Math Help - No trigonometry use? parallelogram

  1. #1
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    No trigonometry use? parallelogram

    In parallelogram ABCD, angle A is acute and AB = 5. Point E is on AD with AE = 4 and BE = 3. A line through B,
    perpendicular to CD, intersects CD at F. If BF = 5, find EF. A geometric solution (no trigonometry) is desired.
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  2. #2
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    As I read it, your parallelogram is not valid. This is because ABF should form a right isosceles triangle, yet BAE is a 5-4-3 triangle (so the angle BAE is less than BAF). Thus, F cannot lie on the line segment CD.

    Perhaps you mean that F lies on the line determined by CD. In this case, one can find EF by simply using the pythagorean theorem and scaling relations. Here is what I imagine your description to be:

    No trigonometry use? parallelogram-parallelogram.png

    I donít quite know exactly what you mean by no trigonometry, but the rest is simply solving and applying 5-4-3 triangles (or more generally, it is a simple scaling relationship to the triangle BAE). First, we see BC is the hypotenuse of BCF with short leg BF, so BC=5/3*BF (thus BC=25/3=AD).
    Then, notice that HBE has short leg EH and long leg BE, so EH=3/4*EB (in fact, EH=9/4). This gives us HD=AD-AE-EH (so HD=25/12).
    HD is the hypotenuse of HFD with short leg HF, so HF=3/5*HD (HF=5/4).
    The last triangle is HFD with hypotenuse HF, so the two legs are HG=3/5*HF (HG=3/4) and FG=4/5*HF (FG=1). Now we can get EG=EH+HG (EG=9).
    Finally, using the Pythagorean theorem, EF^2=EG^2+FG^2 ( EF=\sqrt{10})
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