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Math Help - Circle geometry, finding length of a tangent

  1. #1
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    Circle geometry, finding length of a tangent

    Two circles of radii 5cm and 8cm touch each other externelly, calculate the length of the common tangent.

    and

    if the radii of two intersecting circles are 51cm and 74cm and the length of their common chord is 48cm
    A) Calculate the length of the line joining their centres
    B) Calculate the length of the common tangent



    In both questions i get them all set up draw them out right, i just can't seem to figure out what the common tangent length is (mathematically and literally) maybe there is a rule i am forgetting?

    thanks for the help
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  2. #2
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    Quote Originally Posted by JordanW View Post
    Two circles of radii 5cm and 8cm touch each other externelly, calculate the length of the common tangent.

    ...
    B) Calculate the length of the common tangent



    In both questions i get them all set up draw them out right, i just can't seem to figure out what the common tangent length is (mathematically and literally) maybe there is a rule i am forgetting?

    thanks for the help
    I've attached a sketch.

    The triangle which contains the length of the common tangent, is a right triangle.

    The hypotenuse is R + r

    The short slanted leg is R - r

    Thus the length of the common tangent is:

    l_t=\sqrt{(R+r)^2-(R-r)^2} = \sqrt{R^2+2rR +r^2-(R^2-2rR +r^2)} = 2\sqrt{rR}
    Attached Thumbnails Attached Thumbnails Circle geometry, finding length of a tangent-aeusseregemeins_tang.png  
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  3. #3
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    Hello, JordanW!

    Two circles of radii 5cm and 8cm touch each other externally.
    Calculate the length of the common tangent.
    Code:
                                                      o Q
                                                  *   |
                  * * *               8     *         |
              *           *             *             |
            *               *       *                 |3
           *                 *  *                     |
                        5   *                         |
          *         P   *      *                      |
          *         o - - - - - - - - - - - - - - - - * C
          *         |         *   *                   |
                    |                                 |
           *       5|        *                        |5
            *       |       *          *              |
              *     |     *                 *         |
                  * o * - - - - - - - - - - - - - - * o
                    A                                 B

    The centers of the circles are P and Q.
    Their common tangent is AB.
    PA = 5,\;QB = 8

    Through P, draw PC \parallel AB.
    Then: . CB = 5,\;QC = 3

    In right triangle QCP\!:\;\;PC^2 + QC^2 \:=\:PQ^2 \quad\Rightarrow\quad PC^2 + 3^2\:=\:13^2

    Hence: . PC^2 \:=\:169 - 9 \:=\:160 \quad\Rightarrow\quad PC \:=\:4\sqrt{10}

    Since AB = PC,\;\;AB \:=\:4\sqrt{10}

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  4. #4
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    Quote Originally Posted by JordanW View Post
    ...

    if the radii of two intersecting circles are 51cm and 74cm and the length of their common chord is 48cm
    A) Calculate the length of the line joining their centres
    ...
    Draw a sketch.

    Let c denote half of the common chord. c is simultaneously the leg of two right triangles where you know the length of the hypotenuses.

    Therefore

    |\overline{M_1 M_2}| = \sqrt{R^2-c^2} + \sqrt{r^2-c^2}

    With your values:

    |\overline{M_1 M_2}| = 70 + 45 = 115

    Now calculate the length of the common tangent.
    Attached Thumbnails Attached Thumbnails Circle geometry, finding length of a tangent-zweikrs_gemeinssehne.png  
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