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
OA A
Source: e-GMAT
A bigger circle (with center A) and a smaller circle (with c
This topic has expert replies
-
- Moderator
- Posts: 7187
- Joined: Thu Sep 07, 2017 4:43 pm
- Followed by:23 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
- fskilnik@GMATH
- GMAT Instructor
- Posts: 1449
- Joined: Sat Oct 09, 2010 2:16 pm
- Thanked: 59 times
- Followed by:33 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
\[? = ST\]BTGmoderatorDC 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
Source: e-GMAT
(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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br