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Are all the angles of Triangle ABC smaller than 90 degrees?

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Are all the angles of Triangle ABC smaller than 90 degrees? [#permalink] New post 21 Feb 2007, 03:43
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Are all the angles of Triangle ABC smaller than 90 degrees?

1. 2 * AB = 3 * BC = 4 * AC

2. AC^2 + AB^2 > BC^2
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 [#permalink] New post 21 Feb 2007, 08:43
(A) For me :)

From 1
Such a rule as 2 * AB = 3 * BC = 4 * AC defines a family of similar triangles. Among similar triangles, angles are preserved.

Thus, we can choose 1 case to conclude. AB = 6, BC = 4 and AC = 3.

If we notice that, 6 5 3 is very closed to a right triangle 5 4 3. If we increase from 5 to 6, the angle of C increase above 90.

So, the answer is no

SUFF.

From 2
AC^2 + AB^2 > BC^2

o If AB^2 + BC^2 = AC^2, then we have a right triangle
o If AB=AC=BC=1, then we have an equilateral triangle and all angles are equal to 60.

INSUFF.
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DS- Triangle and Angles [#permalink] New post 10 Nov 2007, 16:05
Are all the angles of triangle ABC smaller than 90 degrees?

1) 2AB=3BC=4AC
2)AC^2+AB^2>BC^2
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Re: DS- Triangle and Angles [#permalink] New post 10 Nov 2007, 16:10
GMATBLACKBELT wrote:
Are all the angles of triangle ABC smaller than 90 degrees?

1) 2AB=3BC=4AC
2)AC^2+AB^2>BC^2



Heres what I did.

Drew a triangle made all sides in terms of AC. Then I said AC= X.

So well have sides 2X, 4X/3, and X

Now since X is the smallest side it gets the smallest angle. So I said this angle corresponding to X = Y.

So I said we have angles Y, 4Y/3, and 2Y.

then Y+4/3Y+2Y=180 Y= ~41.5

So from this we know that no angle is bigger than 90degrees. Suff.


IS THIS CORRECT?


S2: is insuff.
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 [#permalink] New post 10 Nov 2007, 17:35
Dear Gmatblackbelt,

I think you are absolutely correct. one of the properties of triangle is that the angles opposite the the sides of a triangle will be in the same proportion as their sides are.

Answer should be A...

B is insufficient
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Re: DS- Triangle and Angles [#permalink] New post 10 Nov 2007, 20:31
GMATBLACKBELT wrote:
Are all the angles of triangle ABC smaller than 90 degrees?

1) 2AB=3BC=4AC
2)AC^2+AB^2>BC^2


This is an interesting question.

Statement 1 tells us that the sides are in the ratio of 2:3:4. This is sufficient to tell us that the angles are less than 90 degress. How? Simple, use the ratio concept:
2/9*180=40
3/9*180=60
4/9*180=80

Statement 2 tells us that the squares of 2 sides is greater than the square of the 3rd side. This indicates that there is a side which causes an angle greater than 90 degrees, while the other 2 are less than 90. Since the question is asking if all the angles are less than 90, this statement is insufficient to answer the question.

So, the answer is A.
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Re: DS- Triangle and Angles [#permalink] New post 10 Nov 2007, 20:47
Vemuri wrote:
GMATBLACKBELT wrote:
Are all the angles of triangle ABC smaller than 90 degrees?

1) 2AB=3BC=4AC
2)AC^2+AB^2>BC^2


This is an interesting question.

Statement 1 tells us that the sides are in the ratio of 2:3:4. This is sufficient to tell us that the angles are less than 90 degress. How? Simple, use the ratio concept:
2/9*180=40
3/9*180=60
4/9*180=80

Statement 2 tells us that the squares of 2 sides is greater than the square of the 3rd side. This indicates that there is a side which causes an angle greater than 90 degrees, while the other 2 are less than 90. Since the question is asking if all the angles are less than 90, this statement is insufficient to answer the question.

So, the answer is A.


On second thought, since I was able to say in statement 2 that NOT all angles are less than 90, isn't it sufficient to answer the question? Is the answer A or D? I am now confused.
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Re: DS- Triangle and Angles [#permalink] New post 10 Nov 2007, 21:22
Vemuri wrote:
GMATBLACKBELT wrote:
Are all the angles of triangle ABC smaller than 90 degrees?

1) 2AB=3BC=4AC
2)AC^2+AB^2>BC^2


This is an interesting question.

Statement 1 tells us that the sides are in the ratio of 2:3:4. This is sufficient to tell us that the angles are less than 90 degress. How? Simple, use the ratio concept:
2/9*180=40
3/9*180=60
4/9*180=80

Statement 2 tells us that the squares of 2 sides is greater than the square of the 3rd side. This indicates that there is a side which causes an angle greater than 90 degrees, while the other 2 are less than 90. Since the question is asking if all the angles are less than 90, this statement is insufficient to answer the question.

So, the answer is A.


The answer is A, but the reasons are a little different.

I. You can't assume the angles of a triangle are in the same ratio as the sides. Sometimes that's the case (equilateral triangle), but sometimes not (30-60-90 triangle).

FIrst, let's find the ratio of the sides to one another:

2AB = 3 BC, so AB : BC = 3 : 2. 3BC = 4AC, so BC : AC = 4 : 3. To put them all together, multiply AB : BC by 2:

AB : BC : AC = 6 : 4 : 3. If the ratio were 5 : 4 : 3, we'd know that C was a right angle. Since AB is always proportionally longer, we know that C must always be > 90 degrees. SUFFICIENT.

II. Notice that, as the question is worded, there's no guarantee that BC is the longest side of the triangle. So consider these two examples:

a. ABC is equilateral: AC^2 + AB^2 > BC^2.

b. ABC is a right triangle, with C the right angle (so that AB is the hypoteneuse): AC^2 + AB^2 > BC^2.

Each of these cases is consistent with statement II, but they yield different answers for the question, are all angles < 90. So, INSUFFICIENT.
Re: DS- Triangle and Angles   [#permalink] 10 Nov 2007, 21:22
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