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Updated on: 21 May 2020, 01:27
Originally posted by EgmatQuantExpert on 22 Nov 2016, 05:33.
Last edited by Bunuel on 21 May 2020, 01:27, edited 3 times in total.
Last edited by Bunuel on 21 May 2020, 01:27, edited 3 times in total.
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A
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Common Mistakes in Geometry Questions - Exercise Question #4
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
Answer Choices
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Common Mistakes in Geometry are explained in detail in the following post:
Common Mistakes in Geometry Questions
Detailed solution will be posted soon.
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
Answer Choices
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
Common Mistakes in Geometry are explained in detail in the following post:
Common Mistakes in Geometry Questions
Detailed solution will be posted soon.
Updated on: 01 Dec 2016, 01:58
Originally posted by EgmatQuantExpert on 22 Nov 2016, 11:11.
Last edited by EgmatQuantExpert on 01 Dec 2016, 01:58, edited 1 time in total.
Last edited by EgmatQuantExpert on 01 Dec 2016, 01:58, edited 1 time in total.
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Note: This questions is related to the article on Common Errors in Geometry
Kindly go through the article once, before solving the question or going through the solution.
Official Solution
Steps 1 & 2: Understand Question and Draw Inferences
Given that PT is a tangent to the small circle and PS is a tangent to the big circle.
ΔPTB and ΔPSA are right angled at T and S respectively.
We need to find the length of ST.
Step 3: Analyze Statement 1
(1) The radii of the bigger and the smaller circles are 9 cm and 4 cm respectively.
If we drop a perpendicular BD on side AS, we get:
BDST is a rectangle (since all angles of this quadrilateral are right angles).
Therefore, since opposite sides of a rectangle are equal, SD = BT = 4 cm
This means, AD = 9 – 4 = 5cm
In right triangle ADB, by applying Pythagoras Theorem, we get:
B\(D^2\) + A\(D^2\) = A\(B^2\)
That is, B\(D^2\) = (9+4\()^2\) – (5\()^2\)
B\(D^2\) = (13+5)(13-5)
Therefore, BD = 12
Since opposite sides of a rectangle are equal, ST = BD = 12 cm
Since we have been able to determine a unique length of ST, Statement (1) is sufficient.
Step 4: Analyze Statement 2
(2) PB = 52/5 cm
In right triangle BTP, we know the length of only one side: BP.
In order to find the lengths of the other sides of this triangle, we need one more piece of information – either one of the two unknown angles, or one of the two unknown sides.
Since we don’t have this information, we will not be able to find the lengths of the unknown sides of triangle BTP.
Due to a similar reasoning, we will not be able to find the length of the sides of triangle ASP.
So, we will not have enough information to find the length of ST.
Therefore statement 2 is not sufficient to arrive at a unique answer.
Since we got the answer from the first statement, Correct Answer: A
Thanks,
Saquib
Quant Expert
e-GMAT
Kindly go through the article once, before solving the question or going through the solution.
Official Solution
Steps 1 & 2: Understand Question and Draw Inferences
Given that PT is a tangent to the small circle and PS is a tangent to the big circle.
ΔPTB and ΔPSA are right angled at T and S respectively.
We need to find the length of ST.
Step 3: Analyze Statement 1
(1) The radii of the bigger and the smaller circles are 9 cm and 4 cm respectively.
If we drop a perpendicular BD on side AS, we get:
BDST is a rectangle (since all angles of this quadrilateral are right angles).
Therefore, since opposite sides of a rectangle are equal, SD = BT = 4 cm
This means, AD = 9 – 4 = 5cm
In right triangle ADB, by applying Pythagoras Theorem, we get:
B\(D^2\) + A\(D^2\) = A\(B^2\)
That is, B\(D^2\) = (9+4\()^2\) – (5\()^2\)
B\(D^2\) = (13+5)(13-5)
Therefore, BD = 12
Since opposite sides of a rectangle are equal, ST = BD = 12 cm
Since we have been able to determine a unique length of ST, Statement (1) is sufficient.
Step 4: Analyze Statement 2
(2) PB = 52/5 cm
In right triangle BTP, we know the length of only one side: BP.
In order to find the lengths of the other sides of this triangle, we need one more piece of information – either one of the two unknown angles, or one of the two unknown sides.
Since we don’t have this information, we will not be able to find the lengths of the unknown sides of triangle BTP.
Due to a similar reasoning, we will not be able to find the length of the sides of triangle ASP.
So, we will not have enough information to find the length of ST.
Therefore statement 2 is not sufficient to arrive at a unique answer.
Since we got the answer from the first statement, Correct Answer: A
Thanks,
Saquib
Quant Expert
e-GMAT
13 Sep 2018, 18:37
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EgmatQuantExpert
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) The radii of the bigger and the smaller circles are 9 cm and 4 cm respectively
(2) PB = 52/5 cm
(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.
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