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In the figure above, acr QRO is a semicircle. What is the

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In the figure above, acr QRO is a semicircle. What is the

by Jay@ManhattanReview » Mon Dec 10, 2018 12:34 am

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In the figure above, acr QRO is a semicircle. What is the area of the circle with center O?

1) The area of the triangle PQO is 30.
2) The length of QRO is 2.5Ï€.

OA C.
We have to get the area of the circle with center O.

Let's take each statement one by one.

1) The area of the triangle PQO is 30.

1/2 * OP * OQ = 30 => R*2r = 60 => R*r = 30, wether OP = R = radius of the bigger circle and r = radius of the smaller circle

We know that the area of the bigger circle = πR^2, but we can't get the unique value of R. Insufficient.

2) The length of QRO is 2.5Ï€.

=> 2Ï€r/2 = 2.5Ï€ => r = 2.5. We can't get the unique value of R. Insufficient.

(1) and (2) together

From (1), we have R*r = 30 and from (2), we have r = 2.5, thus, R = 30/2.5 = 12.

Thus, the area of the bigger circle = πR^2 = π*(12)^2 = 144π. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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Last edited by Jay@ManhattanReview on Tue Jan 15, 2019 8:41 pm, edited 1 time in total.
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by fskilnik@GMATH » Mon Dec 10, 2018 4:26 am

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In the figure above, arc QRO is a semicircle. What is the area of the circle with center O?

1) The area of the triangle PQO is 30.
2) The length of QRO is 2.5Ï€.
$$? = {S_{{\text{big}}\,\, \odot }}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\boxed{\,? = OP\,}\,$$

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$$\left( {1 + 2} \right)\,\,\,\,\left\{ \matrix{
\,\left( 2 \right)\,\,\, \Rightarrow \,\,\,OQ = 5 \hfill \cr
\,\left( 1 \right)\,\,\, \Rightarrow \,\,\,30 = {{OP \cdot OQ} \over 2} \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,?\,\, = \,\,\,OP\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left( {\rm{C}} \right)$$


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
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