Is triangle ABC an acute angled triangle? 1) [m]c^2 < a^

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Wed Oct 31, 2018 6:50 am

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Is triangle ABC an acute angled triangle?

1) [m]c^2 < a^2 + b^2[/m] where a, b and c are sides of triangle ABC
2) Sides of the triangle ABC have lengths 5, 6 and 7

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Source: Beat The GMAT — Data Sufficiency | Wed Oct 31, 2018 6:50 am

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

by fskilnik@GMATH » Wed Oct 31, 2018 12:53 pm

GMATinsight wrote:Is triangle ABC an acute angled triangle?

1) [m]c^2 < a^2 + b^2[/m] where a, b and c are sides of triangle ABC
2) Sides of the triangle ABC have lengths 5, 6 and 7

Source: www.GMATinsight.com

All angles are measured in degrees.
$${\rm{all}}\,\,\Delta ABC\,\,{\rm{internal}}\,\,{\rm{angles}}\,\,\mathop < \limits^? \,\,\,{90}$$

(We assume c is the length of the side that is opposite to the internal angle ACB, etc.)

$$\left( 1 \right)\,\,\,\,{c^2} < {a^2} + {b^2}\,\,\,\, \Rightarrow \,\,\,\,\,\angle ACB < 90\,\,\,\, \Rightarrow \,\,\,\,{\rm{INSUFF}}.\,$$

$$\left( 2 \right)\,\,\,5,6,7\,\,\,\,\, \Rightarrow \,\,\,\,\,\Delta ABC\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\Delta ABC\,\,{\rm{internal}}\,\,{\rm{angles}}\,\,{\rm{are}}\,\,{\rm{uniquely}}\,\,{\rm{known}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\,\,\,\,$$

POST-MORTEM:
$${7^2} < {5^2} + {6^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\max \,\,\,\Delta ABC\,\,{\rm{internal}}\,\,{\rm{angle}}\,\,\,\, < \,\,90\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$

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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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Wed Oct 31, 2018 11:51 pm

GMATinsight wrote:Is triangle ABC an acute-angled triangle?

1) c^2 < a^2 + b^2, where a, b and c are sides of triangle ABC
2) Sides of the triangle ABC have lengths 5, 6 and 7

Source: www.GMATinsight.com

Let's take each statement one by one.

1) c^2 < a^2 + b^2, where a, b and c are sides of triangle ABC.

=> /_ACB < 90Âº and if /_ABC < 90Âº and /_BAC < 90Âº, the answer is Yes.

However, if a^2 â‰¥ c^2 + b^2, we have /_BAC â‰¥ 90Âº, the answer is No.

No unique answer. Insufficient.

2) Sides of the triangle ABC have lengths 5, 6 and 7.

Given the sides 5, 6, and 7, we know that the largest of the angles would be the one opposite the side with length 7.

Since 5^2 + 6^2 < 7^2 => 25 + 36 < 49 => 61 < 49, we have the largest angle of ABC < 90Âº, thus, ABC is an acute angled triangle.

Sufficient.

The correct answer: B

Hope this helps!

-Jay
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