Tuesday, September 13, 2022

Geometry Problem 1496: Triangle, Congruence, Isosceles, Double Angle, 30 Degrees

Geometry Problem 1496. Post your solution in the comment box below.
Level: Mathematics Education, K-12 School, Honors Geometry, College.

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Geometry Problem 1496: Triangle, Congruence, Isosceles, Double Angle, 30 Degrees

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4 comments:

  1. Trigonometry Solution

    <BCD = <CBD = @ - x
    <ADC = < ACD = @ + x

    Let BD = CD = d

    Now write the Sine Formula for Tr.s BCD, ACD, ABC noting that sin (180 - p) = sin p and that sin 2q = 2.sin q. cos q

    a/d = sin [2(@ - x)] / sin(@ - x) = 2.cos(@ - x).........(1)
    d/b = sin [2(@+x)] / sin(@-x) = 2.cos(@+x)............(2)
    a/b = sin 3@ / sin @ ................................................(3)

    (1) X (2) ;
    a/b = 4.cos (@-x).cos (@+x) = 4.(cos @.cos x + sin @. sin x).(cos @.cos x - sin @.sin x)
    a/b = 4.(cos^2 @. cos^2 x - sin^2 @. sin^2 x) (difference of 2 squares)
    = 4.[(1 - sin^2 @). (1 - sin^2 x) - sin^2 @. sin^2 x) since sin^2 w + cos^2 w = 1
    = 4. (1 - sin^2 @ - sin^2 x)
    = 3 - 4.sin^2 @ from (3)

    So 4. sin^2 x = 1
    and sin x = 1/2 and x = 30

    Sumith Peiris
    Moratuwa
    Sri Lanka

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  2. Geometric Solution:
    Let point E be the middle point of BC. Since BD = CD, therefore ED is the perpendicular bisector of BC.
    Extend ED and let it intersect AB at point F. Hence BF = CF and <FBC = <FCB = \alpha.
    Therefore triangulars ABC and ACF are similar, and AB * AF = AC^2.
    Since AD = AC, thus AB * AF = AD^2, therefore triangulars ABD and ADF are similar, and <ADF = <ABD = x.
    Finally, since <EDC = 90 - \alpha + x, and <CDA = \alpha + x, and <ADF = x,
    therefore, 180 = <EDF = <EDC + <CDA + <ADF = 90 + 3*x, and we conclude x = 30.
    Q.E.D.

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  3. Very good. I got as far as BF=BC but failed to realize that <ADF =x

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  4. Take E on BC so that BE=AE=AC, <AEB=180-2a, take then reflection of E about AB, see that ACBF is an isosceles trapezoid and BF=AF=DF=AD, triangle ADF is equilateral, F is the circumcenter of triangle ADF, thus the required angle is 30 degs!

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