EgmatQuantExpert wrote:

A bigger circle (with center A) and a smaller circle (with center B) are touching each other externally. PT and PS are the tangents drawn to these circles from an external point (as shown in the figure). What is the length of ST?

(1) The radii of the bigger and the smaller circles are 9 cm and 4 cm respectively

(2) PB = 52/5 cm

\(? = ST\)

(1) Sufficient. Please follow the arguments below looking at the image attached.

\(\Delta PTB\,\, \cong \,\,\,\Delta PSA\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \begin{gathered}

\,\frac{4}{9} = \,\frac{{4 + {\text{aux}}}}{{9 + 4 + 4 + {\text{aux}}}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{aux}}\,\,\,{\text{unique}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,PT\,\,\,{\text{unique}} \hfill \\

\,\frac{9}{4} = \frac{{ST + PT}}{{PT}}\,\,\,\,\,\mathop \Rightarrow \limits^{PT\,\,{\text{unique}}} \,\,\,\,?\,\, = \,\,ST\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{SUFF}}. \hfill \\

\end{gathered} \right.\)

\(\left( * \right)\,\,\,\Delta PTB\,\,\,\left\{ \begin{gathered}

TB = 4 \hfill \\

\left( {{\text{4}}\,{\text{ + }}\,{\text{aux}}} \right)\,\,{\text{unique}} \hfill \\

\end{gathered} \right.\,\,\,\,\mathop \Rightarrow \limits^{{\text{Pythagoras}}} \,\,\,\,\,PT\,\,\,\,{\text{unique}}\)

(2) Insufficient. We present the GEOMETRIC BIFURCATION in the image attached.

This solution follows the notations and rationale taught in the GMATH method.

Regards,

Fabio.

Attachments

13_Set18_7t.gif [ 61.78 KiB | Viewed 956 times ]

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Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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