Friday, May 11, 2012

Problem 752: Square, Circle, Tangent, Chord, Angle

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the problem 752 details.

Online Geometry Problem 752: Problem 752: Square, Circle, Tangent, Chord, Angle.

6 comments:

  1. http://img526.imageshack.us/img526/9787/problem752.png
    Connect CE and CF
    Note that CE perpendicular to BE and CF perpendicular to FD
    ∠CDF=∠ECG=25 and ∠DCF=65=> ECF is a right isosceles triangle
    In triangle ECG, external angle x= ∠CEG+∠ECG= 70

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  2. Triangle DCF and ACE are congruent triangles.
    Hence Angle CDF = 25.
    Let x be angle DFG, then angle ECF = 2x -> angle ECG = 2x - 55. Then angle BCD = 90 = 55 + (2x -55 ) Therefore x = 45. By external angle, CGF = 45 + 25 = 70

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  3. Join CE, CF
    Rt ∆BEC ≡ Rt ∆DFC (∵ BC = CD, CE = CF)
    ∴∠CDF = ∠CBE = 25°
    ∠ECD = ∠BCD - ∠BCE = 90° - ∠ECB = 25°
    ∠DCF = 90° - ∠CDF = 90°- 25° = 65°
    Adding: ∠ECF = 90°
    CE = CF implies ∠CFE = ∠CEF = 45°
    ∴ ∠EFD = 45° (∵ CF ⊥ DF)
    Hence x = ∠CDF + ∠EFD = 25° + 45° = 70°

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  4. Right triangle CFD is formed by rotating right triangle CEB about C for 90 deg,
    So ECF is a right isoceles triangle.
    x = 45 deg. + angle ECG = 70 deg.

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  5. http://www.mathematica.gr/forum/viewtopic.php?f=20&t=27270&p=133540

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  6. Tr.s BCE and DCF are congruent so < CDF = 25 =< ECF. Also< DCF = 65

    Hence Tr. ECF is an isoceles right Tr. and so x=70

    Sumith Peiris
    Moratuwa
    Sri Lanka

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