BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Triangle ABC is equilateral and inscribed in the circle with

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

Triangle ABC is equilateral and inscribed in the circle with

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

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

GMATH practice exercise (Quant Class 15)

Image

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)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
Top
Source: — Data Sufficiency |
User avatar
GMAT Instructor
Posts: 1449
Joined: Sat Oct 09, 2010 2:16 pm
Thanked: 59 times
Followed by:33 members

Triangle ABC is equilateral and inscribed in the circle with

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

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

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

Image

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
Image

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

(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
Top
Post new topic Post Reply
• Page 1 of 1