Results 1 to 6 of 6

Math Help - three circles inscribed in a big circle

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    4

    three circles inscribed in a big circle

    hello ! help me to solve this problem, please!!
    its really hard for me ! thank you so much and God bless you richly !

    Three circles A,B and C are tangent externally to each other and each tangent internally to a larger circle having a radius of 10 cm. Radius of circle A is 5 cm.


    a) compute the distance from the center of the larger circle to the point of tangency of the two circles B and C which are identical.

    b) compute the radius of circles B and C.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Hi arenkun,

    For the radii, you can use a combination of Sin(angle) and the Cosine Rule
    (Law of Cosines) to solve this.

    In the diagram...

    \vartriangle{oc_1c_2} has sides 5, 5+r, 10-r.

    Also \sin\theta = \frac{r}{5+r}

    From the Law of Cosines...

    (10-r)^2=(5+r)^2+5^2-2(5)(5+r)\cos\theta

    100-20r+r^2=25+10r+r^2+25-10(5+r)\cos\theta

    50-30r=-(50+10r)\cos\theta

    \cos\theta = \frac{50-30r}{-(50+10r)}=\frac{30r-50}{50+10r}

    Now use the trigonometric identity Sin^{2}\theta + Cos^{2}\theta = 1.

    \frac{(30r-50)^2}{(50+10r)^2}+\frac{r^2}{(5+r)^2} = 1

    \frac{900r^2-3000r+2500}{2500+100r^2+1000r} + \frac{r^2}{25+10r+r^2}=1

    \frac{9r^2-30r+25}{25+r^2+10r}+\frac{r^2}{25+10r+r^2}=1

    10r^2-30r+25=r^2+10r+25

    9r^2-40r=0

    r(9r-40)=0

    As r is not zero, then 9r=40

    r=\frac{40}{9}
    Attached Thumbnails Attached Thumbnails three circles inscribed in a big circle-tangential-circles.jpg  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Once you have "r", you can use it to calculate the answer
    to the first question as shown on the attachment.
    Attached Thumbnails Attached Thumbnails three circles inscribed in a big circle-tangential-circles2.jpg  
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    However, you may have meant the large circle to be the green one!

    Let us know if you did.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jan 2010
    Posts
    4
    TO : Archie Meade


    THANK YOU so much, Archie Meade !
    it's really a big help to me..
    May God bless you more.
    I'm so happy. thanks again for the idea.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    You're more than welcome, arenkun,
    thank you too.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inscribed circles in MuPad
    Posted in the Math Software Forum
    Replies: 1
    Last Post: September 15th 2011, 04:37 PM
  2. Hyperbolic triangles and inscribed circles
    Posted in the Geometry Forum
    Replies: 0
    Last Post: April 8th 2010, 01:36 PM
  3. Triangle inscribed in two circles...
    Posted in the Geometry Forum
    Replies: 3
    Last Post: February 8th 2010, 10:10 PM
  4. Replies: 2
    Last Post: February 6th 2010, 09:31 AM
  5. Three Inscribed Circles Into Big Circle
    Posted in the Geometry Forum
    Replies: 5
    Last Post: August 20th 2009, 02:03 PM

Search Tags


/mathhelpforum @mathhelpforum