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Two lines PA and PB are tangent lines to the circle. What is the measure of

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by Max@Math Revolution » Fri Aug 07, 2020 12:06 am

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[GMAT math practice question]

Two lines PA and PB are tangent lines to the circle. What is the measure of \(<\) x?
8.7ds.png
1) \(<\) AOB = 108.
2) \(<\) APB = 72.
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Source: — Data Sufficiency |
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Re: Two lines PA and PB are tangent lines to the circle. What is the measure of

by Max@Math Revolution » Sun Aug 09, 2020 11:02 pm
=>
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question.
As shown in the figure below, if we know the measure of \(<\) AOB, we can find the measure of \(<\) x, since \(<\) x =\(<\) AOB/2.

Let’s look at each condition separately.

Condition (1) tells us that \(<\) AOB = 108, from which, since \(<\) x = \(<\)AOB/2, we get \(<\) x = \(<\)AOB/2, \(<\) x = 108/2 = 54.

The answer is unique, yes, so the condition is sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
8.7PS(A).png
Condition (2) tells us that \(<\) APB = 72. Then, since \(<\) APB + \(<\) AOB = 180, if we substitute \(<\) APB = 72 into this equation, we get 72 + \(<\) AOB = 180, or \(<\) AOB = 180 - 72 = 108, which is equal to condition (1). So, as shown above, this condition is also sufficient.

Also, the value of condition (1) is equal to the value of condition (2), so by Tip 1, we get D as the most likely answer.

Each condition alone is sufficient.
Therefore, D is the correct answer.
Answer: D
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