Circumference

Dec 2008
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Triangle ABC is circumscribed about cirlce O. P and Q are points of tangency such that BP=3 and CQ=5. Waht is the measure of BC?
 

dwsmith

MHF Hall of Honor
Mar 2010
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582
Florida
Triangle ABC is circumscribed about cirlce O. P and Q are points of tangency such that BP=3 and CQ=5. Waht is the measure of BC?
Have you drawn the picture?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
Triangle ABC is circumscribed about cirlce O. P and Q are points of tangency such that BP=3 and CQ=5. Waht is the measure of BC?
P and Q are tangent points. These points can't be on the circle then because if triangle ABC is inscribed inside and the points are on the circle, then they are secant points. Where did you put these points on the drawing?
 

Soroban

MHF Hall of Honor
May 2006
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Lexington, MA (USA)
Hello, winsome!

Triangle ABC is circumscribed about cirlce O.
P and Q are points of tangency such that BP = 3 and CQ = 5.
What is the measure of BC?
Code:
                        A
                        o
                       / \
                      /   \
                     /     \
                    /       \
                   /         \
                  /   * * *   \
                 /*           *\
                *               *
             P o                 o Q
              /                   \
             /*                   *\
            / *         *         * \
         3 /  *         O         *  \ 5
          /                           \
         /     *                 *     \
        /       *               *       \ 
       /          *            *         \
    B o - - - - - - - * o * - - - - - - - o C
              3         R         5

Side \(\displaystyle AB\) is tangent to the circle at \(\displaystyle P.\)
Side \(\displaystyle AC\) is tangent to the circle at \(\displaystyle Q.\)
Side \(\displaystyle BC\) is tangent to the circle at \(\displaystyle R.\)

We are given: .\(\displaystyle BP \,=\,3,\;CQ \,=\,5.\)


The tangents drawn from an external point to a circle are equal.

. . Hence: .\(\displaystyle BR = 3,\;CR = 5.\)


Therefore: .\(\displaystyle BC \;=\;3 + 5 \;=\;8\)

 
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