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# Is triangle ABC an acute angled triangle? 1) [m]c^2 < a^2 + b^2[/m] w

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Is triangle ABC an acute angled triangle? 1) [m]c^2 < a^2 + b^2[/m] w  [#permalink]

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31 Oct 2018, 07:40
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Difficulty:

85% (hard)

Question Stats:

25% (01:15) correct 75% (01:08) wrong based on 49 sessions

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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

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Is triangle ABC an acute angled triangle? 1) [m]c^2 < a^2 + b^2[/m] w  [#permalink]

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31 Oct 2018, 08:19
2
1

CONCEPT:

If $$a^2+b^2 < c^2$$, it means the angle between side a & b is obtuse, the triangle is obtuse.
If $$a^2+b^2 > c^2$$, it means the angle between a & b is acute, but since other angle can be obtuse, the triangle cant be taken as acute.
If $$a^2+b^2 =c^2$$, it means the angle between a & b is 90 deg, hence the triangle is right.

St1)
$$a^2+b^2 > c^2$$, it means the angle between a & b is acute, but since other angle can be acute/obtuse, the triangle cant be taken as acute.
NOT SUFFICIENT.

St2) Sides of the triangle ABC have lengths 5, 6 and 7
As all the sides are known, a unique triangle can be made and hence angle can be found.
So, we can say whether the triangle is acute or not.
SUFFICIENT.

To understand better, please see sketch
Attachment:

gmatbusters.jpg [ 120.19 KiB | Viewed 482 times ]

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: http://www.GMATinsight.com

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Re: Is triangle ABC an acute angled triangle? 1) [m]c^2 < a^2 + b^2[/m] w  [#permalink]

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31 Oct 2018, 14:06
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: http://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.
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Re: Is triangle ABC an acute angled triangle? 1) [m]c^2 < a^2 + b^2[/m] w  [#permalink]

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01 Nov 2018, 04:18
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: http://www.GMATinsight.com

A question on similar property can be seen here

https://gmatclub.com/forum/in-the-diagr ... l#p2158190

The concept of the acute angled, Right angles and Obtuse angled triangle is as shown in the file attached.

Please pay attention to the fact that here c is representing the longest side of the triangle
Attachments

File comment: www.GMATinsight.com

Geometry 3 Right Acute and Obtuse angle triangle.jpg [ 96.22 KiB | Viewed 403 times ]

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Re: Is triangle ABC an acute angled triangle? 1) [m]c^2 < a^2 + b^2[/m] w   [#permalink] 01 Nov 2018, 04:18
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