Friday, April 23, 2010

Problem 436: Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement.

Proposed Problem
Click the figure below to see the complete problem 436 about Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement.

Problem 436.<br />Tangent Circles, Diameter, Chord, Perpendicular, Congruence, Measurement.
See also:
Complete Problem 436

Level: High School, SAT Prep, College geometry

3 comments:

  1. ▲AFE ~ ▲AGH => AF/AG = AE/AH (right tr & comm ang A)
    ▲AGB ~ ▲AFC => AF/AG = AC/AB (right tr & comm ang A)
    =>
    AE/AH = AC/AB =>
    AB∙AE = AC∙AH =>
    (AE + EB)∙AE = (AH+HC)∙AH
    AE² + EB∙AE = AH² + HC∙AH
    AE² + ED² = AH² + MH² (ED², MH² from euclid Theorem)
    AD² = AM²

    AD = AM
    ---------------------------------------------

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  2. Problem 436
    Is AG^2/AF^2=(AH.AB)/(AE.AC),but AG^2/AF^2=AH^2/AE^2=(AH.AB)/(AE.AC) so AH/AE=AB/AC. But AD^2/AM^2=(AE.AB)/(AH.AC=1.Therefore AD=AM.
    APOSTOLIS MANOLOUDIS 4 HIGH SHCOOL OF KORYDALLOS PIRAEUS GREECE

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  3. By easy angle chase we get tr. ADG ~ tr. AFD and tr. AMG ~ tr. AFM, from their similitude inferring AD^2=AF.AG=AM^2, done

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