Triangle ABC is equilateral and inscribed in the circle with

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Tue Feb 12, 2019 5:49 am

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GMATH practice exercise (Quant Class 15)

Triangle ABC is equilateral and inscribed in the circle with center O. What is the value of the shaded area?

(1) The length of the radius of the circle is 2
(2) Lines PQ and BC are parallel

Answer: [spoiler]____(C)__[/spoiler]

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Source: Beat The GMAT — Data Sufficiency | Tue Feb 12, 2019 5:49 am

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

Triangle ABC is equilateral and inscribed in the circle with

by fskilnik@GMATH » Wed Feb 13, 2019 6:26 am

fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 15)

Triangle ABC is equilateral and inscribed in the circle with center O. What is the value of the shaded area?

(1) The length of the radius of the circle is 2
(2) Lines PQ and BC are parallel

$$? = {S_{\Delta APQ}}$$

(1) Clearly insufficient, as PROVED by the algebraic-geometric bifurcation (figure, first row):

> In the figure on the left, we have chosen P sufficiently close to B such that our focus is "as near as desired" to HALF (say 49.99%) of the area of the triangle ABC.

> In the figure on the right, we have chosen P such that lines PQ and BC are parallel, hence our focus is 4/9 (= 0.4444... , i.e., approx. 44.44%) of the area of the triangle ABC.
(The number 4/9 will be obtained in (1+2) below!)

(2) Trivial geometric bifurcation shown in the figure (second row).

(1+2) Triangles APQ and ABC are similar, hence using the "30-60-90" shortcut we get the last figure, and from it (explain):

$${{{S_{\Delta APQ}}} \over {{S_{\Delta ABC}}}} = {\left( {{2 \over {1 + 2}}} \right)^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {S_{\Delta APQ}} = {4 \over 9}\left( {{S_{\Delta ABC}}} \right)\,\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
Important: there is no need for obtaining the area of triangle ABC explicitly: given the numerical value of the radius of the circle, triangle ABC (equilateral and inscribed in it) is unique, and its unique area COULD be calculated!

The correct answer is (C).

We follow 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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