Results 1 to 5 of 5

Math Help - Circles and Tangents

  1. #1
    Member Mr Rayon's Avatar
    Joined
    Sep 2007
    Posts
    137

    Circles and Tangents

    SEE ATTACHMENT FOR QUESTION DETAILS
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

  2. #2
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    Quote Originally Posted by Bradley View Post
    At last someone who can pose a math question with symbols & pictures without resorting to Latex (I prefer MS Excel myself).
    What's wrong with latex ...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Nov 2007
    Posts
    54

    Sorry, no time for pictures, but here you go.

    Call the horizontal leg of the triangle x-axis.

    Call the vertical leg of the triangle y-axis.

    On the x-axis draw a line perpendicular such that it will be tangent to the circle on the left side.

    On the y-axis draw a line perpendicular such that it will be tangent to the circle on the top.

    Already you know the measurements of the entire x-axis and y-axis, or legs of the triangle.

    It should not be very difficult to measure the diameter of the circle from the box you just drew around it.

    If that is not in the spirit of your text, look up TANGENT TO DIAMETER THEOREM:

    If a chord in a plane of a circle is perpendicular to a tangent at the point of tangency, the chord is a diameter.
    You have at least two points of tangency in your drawing.
    ---------------------------
    Here are instructions from a text:

    At the point of tangency of each tangent line, draw a chord perpendicular to each tangent.
    Because of the TANGENT TO DIAMETER THEOREM, these two chords will be diameters and will intersect each other at the center of the circle.
    ---------------------------
    That means you will also have two right triangles that point to nose-to-nose in the center.
    The base of each triangle will be a radius.
    You are no better off as you will still have to measure.
    If you find a numeric solution, please let me know.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    I've modified your sketch a little bit.

    1. Use Tangens to calculate \angle \alpha = 36.87^\circ

    2. The left right triangle will give you:

    x = (8-x) \cdot \tan\left(\frac{\alpha}{2}\right)

    3. Expand the bracket, collect like terms:

    x+ x \cdot \tan\left(\frac{\alpha}{2}\right) = 8 \cdot  \tan\left(\frac{\alpha}{2}\right)

    and therefore:

    x = \frac{8 \cdot  \tan\left( \frac{\alpha}{2}\right)}{1+\tan\left(\frac{\alpha}  {2}\right)}~\implies~ x\approx 2.0
    Attached Thumbnails Attached Thumbnails Circles and Tangents-inkreisradius_rwd.gif  
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123

    general solution

    With your problem you were looking for the length of the radius of the inscribed circle.

    1. If you know the length of the three sides of a triangle you can calculate
    - the interior angles (Cosine rule)
    - the area of the triangle

    2. The center of the incircle is the intersection of the angle bisectors of the interior angles.

    3. Divide the triangle into 3 triangles so that the radius of the incircle is the height. r_1 = r_2 = r_3 = r

    4. Let A_{total} denote the area of the complete triangle. Then you know:

    A_{total} = \underbrace{\frac12 \cdot c \cdot r}_{red \ area} + \underbrace{\frac12 \cdot a \cdot r}_{blue \ area} + \underbrace{\frac12 \cdot b \cdot r}_{green \ area}

    A_{total}=\frac12 \cdot (a+b+c) \cdot r

    \boxed{r = \frac{2A_{total}}{a+b+c}}

    With your problem:

    a = 8 cm, b = 6 cm, c = 10 cm

    A_{total} = 24 \ cm^2

    And therefore: r = \frac{2 \cdot 24}{6+8+10} = 2
    Attached Thumbnails Attached Thumbnails Circles and Tangents-rho_perflaeche.gif  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 4 circles and 3 tangents
    Posted in the Geometry Forum
    Replies: 1
    Last Post: August 28th 2009, 12:32 AM
  2. Circles and tangents
    Posted in the Geometry Forum
    Replies: 3
    Last Post: July 28th 2008, 10:51 PM
  3. Tangents & Circles
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: April 14th 2008, 06:01 PM
  4. tangents and circles
    Posted in the Geometry Forum
    Replies: 11
    Last Post: January 13th 2008, 03:10 PM
  5. Circles and tangents
    Posted in the Geometry Forum
    Replies: 1
    Last Post: November 11th 2006, 08:17 AM

Search Tags


/mathhelpforum @mathhelpforum