| A circle is touching the side BC of \[\Delta ABC\] at P and is also touching AB and AC produced at Q and R, respectively. If \[AQ=6\,\,cm,\] then perimeter of \[\Delta ABC\] is |
A) \[6\,\,cm\]
B) \[10\,\,cm\]
C) \[12\,\,cm\]
D) \[18\,\,cm\]
Correct Answer: C
Solution :
| We know that, the length of tangents drawn to a circle from any external point are equal. |
 |
| \[\therefore \] \[AQ=AR\] (i) |
| \[BP=BQ\] (ii) |
| and \[CP=CR\] ... (iii) |
| Now. perimeter of \[\Delta ABC=AB+BC+AC\] |
| \[=AB+BP+CP+AC\] |
| Here, \[BQ=BP\] and \[CP=CR\] |
| \[=AB+BQ+CR+AC\] |
| [using Eqs. (ii) and (iii)] |
| \[AB+BQ=AQ\] ... (iv) |
| \[AC+CR=AR\] ... (v) |
| Now, adding Eqs. (iv) and (v) |
| \[AB+BQ+AC+CR=AQ+AR\] |
| [Now, AQ = AR] |
| \[\Rightarrow \] \[AB+BQ+AC+CR=2AQ\] |
| \[=2\times 6=12\] |
| Now, perimeter of \[\Delta ABC=2\times 6=12\] |
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