A circle is inscribed in a ^@ \triangle ABC ^@, touching ^@ BC, CA, ^@ and ^@ AB ^@ at points ^@ P, Q, ^@ and ^@ R ^@ respectively. If ^@ AB = 14 \space cm, AQ = 11 \space cm ^@ and ^@ CQ = 9 \space cm ^@ then find the length of ^@ BC ^@.
R Q P B C A


Answer:

^@ 12 \space cm ^@

Step by Step Explanation:
  1. We know that the lengths of tangents drawn from an external point to a circle are equal.
    Thus, @^ \begin{aligned} & AR = AQ = 11 \space cm \\ & BP = BR \\ \text{ and } & CP = CQ = 9 \space cm \end{aligned} @^
  2. We see that ^@ BR = AB - AR = 14 \space cm - 11 \space cm = 3 \space cm. ^@
    Thus, ^@ BP = BR = 3 \space cm. ^@
  3. Therefore, ^@ BC = BP + CP = 3 \space cm + 9 \space cm = 12 \space cm. ^@

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