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# The sides of a triangle are a, b, and c. Are the three angles all less

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Math Expert
Joined: 02 Sep 2009
Posts: 60687
The sides of a triangle are a, b, and c. Are the three angles all less  [#permalink]

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06 Nov 2019, 02:29
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Difficulty:

75% (hard)

Question Stats:

41% (02:24) correct 59% (02:20) wrong based on 32 sessions

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The sides of a triangle are a, b, and c. Are the three angles all less than 90 measure degrees?

(1) The areas of the semi-circles with the radius a, b, c are 4, 5, 6, respectively.

(2) $$c < a + b < c + 2$$

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Joined: 19 Oct 2018
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Location: India
Re: The sides of a triangle are a, b, and c. Are the three angles all less  [#permalink]

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06 Nov 2019, 05:46
Statement 1

We know the areas of triangle; hence, we can find the radius of the circles.
We know all the sides of the triangle; we can calculate all the angles using cosine formula or we can check whether all the angles are acute or not using pythagorean inequality.

Sufficient

Statement 2-

a+b<c+2

1. if a=1.4; b=1.4; and c=1.4 (it's an equilateral triangle)

2. if a=0.6; b=0.6; and c=0.9 (it's an obtuse angle triangle)
(0.6)^2+(0.6)^2<(0.9)^2
0.72<0.81

Insufficient

Bunuel wrote:
The sides of a triangle are a, b, and c. Are the three angles all less than 90 measure degrees?

(1) The areas of the semi-circles with the radius a, b, c are 4, 5, 6, respectively.

(2) $$c < a + b < c + 2$$

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Manager
Joined: 10 Jun 2019
Posts: 119
Re: The sides of a triangle are a, b, and c. Are the three angles all less  [#permalink]

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20 Nov 2019, 04:34
Statement 1 : From the area of the semi circles we can find the sides.The triangle will have a unique set of sides and hence a unique set of angles .From the angles we will be able to tell if the triangle is acute or isn't.
OR
Once we have the sides, we will perform the acute angle test: that is for all the angles of the triangle to be acute(<90) , the sum of the squares of any two sides MUST be greater than the square of the last side. That is, a^2 +b^2>c^2 and c^2 +b^2>b^2 and a^2 +c^2>b^2 .However, since c is the largest side, we only to need to do it for a^2 +b^2>c^2 .After all, the largest side is opposite the largest angle and if it happens to be acute,the other angles must also be acute

SUFFICIENT

Statement 2 :
If the sides are either 1/1/1( equilateral triangle)or if the sides are 1/1/1.414 (right angled isosceles) the statement holds true.
INSUFFICIENT

Re: The sides of a triangle are a, b, and c. Are the three angles all less   [#permalink] 20 Nov 2019, 04:34
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