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
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/ \
/ \
/ \
/ \
/ \
/ * * * \
/* *\
* *
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: .$\displaystyle BC \;=\;3 + 5 \;=\;8$