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Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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Question Stats: 37% (01:47) correct 63% (01:48) wrong based on 263 sessions

### HideShow timer Statistics Are all angles of triangle ABC smaller than 90 degrees?

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

m17 q04

Originally posted by GMATPASSION on 19 Mar 2012, 02:43.
Last edited by Bunuel on 25 Jun 2013, 02:49, edited 3 times in total.
Edited the question
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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7
8
We don't need to calculate anything for this question.

Are all angles of triangle ABC smaller than 90 degrees?

(1) 2AB = 3BC = 4AC --> we have the ratio of the sides: AB:BC:AC=6:4:3. Now, ALL triangles with this ratio are similar and have the same fixed angles. No matter what these angles actually are, the main point is that we can get them and thus answer the question. Sufficient.

(2) AC^2 + AB^2 > BC^2 --> this condition will hold for equilateral triangle (all angles are 60 degrees) as well as for a right triangle with right angle at B or C (so in this case one angle will be 90 degrees, for example consider a right triangle: AC=5, AB=4 and BC=3). Not sufficient.

Hope it's clear.
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GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 Re: GMAT club Geometry Q  [#permalink]

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2
2
Statement 1: 2AB = 3BC = 4AC
=> AB = 2AC and BC = 4/3 AC
=> AB^2 + BC^2 = 4AC^2 + 16/9 AC^2 = 52 AC^2/9
=> AC^2 > AB^2 + BC^2
=> Angle B is obtuse
=> All angles are not acute. Sufficient.

Statement 2: AC^2 + AB^2 > BC^2
=> Angle A is acute
But this tells us nothing about the other angles. Insufficient.

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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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1st statement can be used to calculate the angles, hence A is the answer.
The second statement doesn't give any precise info.
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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"(1) 2AB = 3BC = 4AC --> we have the ratio of the sides: AB:BC:AC=6:4:3. Now, ALL triangles with this ratio are similar and have the same fixed angles. No matter what these angles actually are, the main point is that we can get them and thus answer the question. Sufficient."

In reference to Bunnel's explanation -

How can we get the angles from the ratio of the sides? Will the angles be in the same ratio as sides?? This is not true for 45-45-90 and 30-60-90 triangles, the angles do not have the same ratio as the sides? How would you compute the actual angles given the ratio of sides, in statement 1?
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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teal wrote:
"(1) 2AB = 3BC = 4AC --> we have the ratio of the sides: AB:BC:AC=6:4:3. Now, ALL triangles with this ratio are similar and have the same fixed angles. No matter what these angles actually are, the main point is that we can get them and thus answer the question. Sufficient."

In reference to Bunnel's explanation -

How can we get the angles from the ratio of the sides? Will the angles be in the same ratio as sides?? This is not true for 45-45-90 and 30-60-90 triangles, the angles do not have the same ratio as the sides? How would you compute the actual angles given the ratio of sides, in statement 1?

Please, take a look at my previous post:

m17-97962.html#p1108349
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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Why is the ratio 6:4:3 for AB:BC:AC? Not seeing where the 6 came from.
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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shirisha091 wrote:
Why is the ratio 6:4:3 for AB:BC:AC? Not seeing where the 6 came from.

Divide through the given equality 2AB = 3BC = 4AC by 12 and get $$\frac{AB}{6}=\frac{BC}{4}=\frac{AC}{3}$$, which can also be written as AB:BC:AC = 6:4:3.
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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(2) AC^2 + AB^2 > BC^2 --> this condition will hold for equilateral triangle (all angles are 60 degrees) as well as for a right triangle with right angle at B or C (so in this case one angle will be 90 degrees, for example consider a right triangle: AC=5, AB=4 and BC=3). Not sufficient.

Bunuel,

Can you please explain the above mentioned statement. I am not able to understand
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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greatps24 wrote:
(2) AC^2 + AB^2 > BC^2 --> this condition will hold for equilateral triangle (all angles are 60 degrees) as well as for a right triangle with right angle at B or C (so in this case one angle will be 90 degrees, for example consider a right triangle: AC=5, AB=4 and BC=3). Not sufficient.

Bunuel,

Can you please explain the above mentioned statement. I am not able to understand

If AC=AB=BC=1 (satisfies AC^2 + AB^2 > BC^2) --> ABC is an equilateral triangle (all angles are 60 degrees) --> all angles are less than 90 degrees.
If AC=5, AB=4 and BC=3 (satisfies AC^2 + AB^2 > BC^2) --> ABC is a right triangle --> NOT all angles are less than 90 degrees.

Hope it's clear.
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Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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Are all angles of triangle ABC smaller than 90 degrees?

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

A triangle where a^2 + b^2 = c^2 is a right triangle, but is there any way of determining if measures greater than or less than that (say, if a^2 + b^2 was greater than c^2) lead to a triangle having measures greater than or less than 90?

I see how 1 is sufficient. ABC will always have that exact ratio of side lengths. The triangle could be 2x3x4 or 200x300x400 but the angle measurements will always be the same.
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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WholeLottaLove wrote:
Are all angles of triangle ABC smaller than 90 degrees?

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

A triangle where a^2 + b^2 = c^2 is a right triangle, but is there any way of determining if measures greater than or less than that (say, if a^2 + b^2 was greater than c^2) lead to a triangle having measures greater than or less than 90?

I see how 1 is sufficient. ABC will always have that exact ratio of side lengths. The triangle could be 2x3x4 or 200x300x400 but the angle measurements will always be the same.

You can see why (2) is not sufficient here: are-all-angles-of-triangle-abc-smaller-than-90-degrees-129298.html#p1060697

As for your other question, say the lengths of the sides of a triangle are a, b, and c, where the largest side is c.

For a right triangle: $$a^2 +b^2= c^2$$.
For an acute triangle: $$a^2 +b^2>c^2$$.
For an obtuse triangle: $$a^2 +b^2<c^2$$.
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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Bunuel wrote:
WholeLottaLove wrote:
Are all angles of triangle ABC smaller than 90 degrees?

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

For a right triangle: $$a^2 +b^2= c^2$$.
For an acute triangle: $$a^2 +b^2>c^2$$.
For an obtuse triangle: $$a^2 +b^2<c^2$$.

Then wouldn't 2) be sufficient? According to $$a^2 +b^2<c^2$$, AC^2 + AB^2 > BC^2 shows that the two legs combined are greater than the third implying that this is an acute triangle.
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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WholeLottaLove wrote:
Bunuel wrote:
WholeLottaLove wrote:
Are all angles of triangle ABC smaller than 90 degrees?

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

For a right triangle: $$a^2 +b^2= c^2$$.
For an acute triangle: $$a^2 +b^2>c^2$$.
For an obtuse triangle: $$a^2 +b^2<c^2$$.

Then wouldn't 2) be sufficient? According to $$a^2 +b^2<c^2$$, AC^2 + AB^2 > BC^2 shows that the two legs combined are greater than the third implying that this is an acute triangle.

No, because we don't know whether BC is the largest side.
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Re: GMAT club Geometry Q  [#permalink]

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GyanOne wrote:
Statement 1: 2AB = 3BC = 4AC
=> AB = 2AC and BC = 4/3 AC
=> AB^2 + BC^2 = 4AC^2 + 16/9 AC^2 = 52 AC^2/9
=> AC^2 > AB^2 + BC^2
=> Angle B is obtuse
=> All angles are not acute. Sufficient.

Statement 2: AC^2 + AB^2 > BC^2
=> Angle A is acute
But this tells us nothing about the other angles. Insufficient.

In the above post I think it should be
AC^2 <AB^2 +BC^2 and not AC^2 > AB^2 + BC^2 as AB^2 + BC^2 = 52 AC^2/9 > AC^2

B is acute

For angle C , taking sides as Shown below

BC= (2/3)AB
AC= (2/4)AB

so BC^2 +AC^2 = (4/9 )AB^2+(1 /4)AB ^2 = (25/36)AB^2
so BC^2 +AC^2 < AB^2 hence C is obtuse

Angle C makes it sufficient not B .

Please do correct , if something still needs attention. Thank you
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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Bunuel, could you explain how we will find angles when we have ratio of sides as in 1). I understand that it will have the same angles, but how can we find them?
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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Ergenekon wrote:
Bunuel, could you explain how we will find angles when we have ratio of sides as in 1). I understand that it will have the same angles, but how can we find them?

Finding the angles is not our aim (and this is not what you need to know for the GMAT). The aim is to determine whether we CAN find them.
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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Bunuel,

a^2 + b^2 < c^2 ------------------ (c is largest then obtuse angle)
so,
3^2 + 4^2 < 6^2 -------------------- Then we have an obtuse angle.

So statement 1 becomes insufficient,
Plz help to clarify....
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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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NickHalden wrote:
Bunuel,

a^2 + b^2 < c^2 ------------------ (c is largest then obtuse angle)
so,
3^2 + 4^2 < 6^2 -------------------- Then we have an obtuse angle.

So statement 1 becomes insufficient,
Plz help to clarify....

The question asks: are all angles of triangle ABC smaller than 90 degrees?

Any triangle which has the ratio of the sides 6:4:3, will be an obtuse triangle, and thus we have an YES answer to the question:
Attachments MSP3420749f5897f2deg2000045fhi8h0c14f9e4f.gif [ 1.32 KiB | Viewed 3858 times ]

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Re: Are all angles of triangle ABC smaller than 90 degrees?  [#permalink]

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Hii
please see it like this: When we don't know the relative lengths of the Sides AB, BC and AC.
$$AC^2 + AB^2 = BC^2$$; it simply states that A is right angle, Hence triangle ABC is right angled triangle.
$$AC^2 + AB^2 < BC^2$$; it states that Angle A is obtuse, hence triangle ABC is obtuse angled triangle.
$$AC^2 + AB^2 > BC^2$$; it only states that Angle A is acute, The triangle ABC can be acute angled triangle or obtuse angled triangle or even right angled triangle.
_________________ Re: Are all angles of triangle ABC smaller than 90 degrees?   [#permalink] 27 Mar 2018, 05:00

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