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?

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/ * * * \
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P o o Q
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3 / * O * \ 5
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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: .\(\displaystyle BC \;=\;3 + 5 \;=\;8\)