Wednesday, October 28, 2015

Geometry Problem 1158: Four Tangent Circles, Tangent Chord, Radius, Metric Relations

Geometry Problem. Post your solution in the comments box below.
Level: Mathematics Education, High School, Honors Geometry, College.

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Online Math: Geometry Problem 1158: Four Tangent Circles, Tangent Chord, Radius, Metric Relations.

Posted by Antonio Gutierrez at 4:50 PM
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2 comments:

  1. Peter TranOctober 28, 2015 at 5:57 PM

    http://s21.postimg.org/5i7u2r6nr/pro_1158.png

    Let x= O4M
    Since O2O3E is a right triangle => O3E^2= 9^2-1^2= 80
    Since O4EO3 is a right triangle => O3O4=Sqrt((x+4)^2+80)
    We have O4C=O4D => x+10= sqrt((x+4)^2+80) +4
    Solve this equation we get x= 15
    In right triangle CBF we have MB^2= MC. MF= 10 .x 40= 400
    So MB=20 and AB=40

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  2. Sumith PeirisOctober 29, 2015 at 8:48 AM

    Let O2O4 meet AB at C the point of tangency and mid point of AB. Let O1D be an altitude of Tr. O1O2O4 and let O1D = h

    We need to use Pythagoras thrice

    h^2 = 9^2 -1 = 80 .... (1) from Tr. O1O2D

    If the radius of O4 = r then
    (r-4)^2 = h^2 + (r-6)^2 from which upon simplifying and using (1) r = 25 (Tr. O1O4D)

    Now use Pythagoras on Tr. ACO4

    (AB/2)^2 = r^2 -(r-10)^2 = 20r-100 = 400

    So AB = 40

    Sumith Peiris
    Moratuwa
    Sri Lanka

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