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M08-31

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M08-31  [#permalink]

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New post 16 Sep 2014, 00:38
1
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A
B
C
D
E

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  85% (hard)

Question Stats:

47% (01:02) correct 53% (01:00) wrong based on 100 sessions

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The angles in a triangle are \(x\), \(3x\), and \(5x\) degrees. If \(a\), \(b\) and \(c\) are the lengths of the sides opposite to angles \(x\), \(3x\), and \(5x\) respectively, then which of the following must be true?

I. \(c \gt a+b\)

II. \(c:a:b=10:6:2\)

III. \(c^2 \gt a^2+b^2\)


A. I and III only
B. II and III only
C. I only
D. II only
E. III only

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Re M08-31  [#permalink]

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New post 16 Sep 2014, 00:38
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Official Solution:


The angles in a triangle are \(x\), \(3x\), and \(5x\) degrees. If \(a\), \(b\) and \(c\) are the lengths of the sides opposite to angles \(x\), \(3x\), and \(5x\) respectively, then which of the following must be true?

I. \(c \gt a+b\)

II. \(c:a:b=10:6:2\)

III. \(c^2 \gt a^2+b^2\)


A. I and III only
B. II and III only
C. I only
D. II only
E. III only


According to the relationship of the sides of a triangle: the length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. Thus I and II can never be true: one side (\(c\)) cannot be larger than the sum of the other two sides (\(a\) and \(b\)). Note that II is basically the same as I: if \(c=10\), \(a=6\) and \(b=2\) then \(c \gt a+b\), which can never be true. Thus even not considering the angles, we can see that only answer choice E (III only) is left (all other options are out because each of them has either I or II in them).

Now, if interested why III is true: as the angles in a triangle are \(x\), \(3x\), and \(5x\) degrees then \(x+3x+5x=180\). Hence \(x=20\), \(3x=60\), and \(5x=100\). Next, if angle opposite side \(c\) were 90 degrees, then according to Pythagoras theorem \(c^2=a^2+b^2\), but since the angle opposite side \(c\) is more than 90 degrees (100) then side \(c\) is larger, hence \(c^2>a^2+b^2\).


Answer: E
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Re: M08-31  [#permalink]

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New post 18 Feb 2016, 02:29
Since C>a-b but not a+b, (1) is wrong. a+b>c on the other hand.
(2) is about ratio of length. Though it might seem that we cannot find length of arms just from angel degrees, but it is ratio not the actual length. So we find the (1) from (2) in fact. So, don't hold.
(3) is right since c is even greater that right angel.
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Re: M08-31  [#permalink]

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New post 20 Feb 2016, 14:22
Bunuel wrote:
The angles in a triangle are \(x\), \(3x\), and \(5x\) degrees. If \(a\), \(b\) and \(c\) are the lengths of the sides opposite to angles \(x\), \(3x\), and \(5x\) respectively, then which of the following must be true?

I. \(c \gt a+b\)

II. \(c:a:b=10:6:2\)

III. \(c^2 \gt a^2+b^2\)


A. I and III only
B. II and III only
C. I only
D. II only
E. III only


we can eliminate right away A and C. in a triangle, the sum of the 2 sides will ALWAYS be greater than the third side.
we are left with B, D, and E.

since we know the angles are in x, 3x, and 5x ratio, the sides must be in the same ratio.
c:a:b=5:1:3 or 10:2:6.
since the order in B is not correct, we can eliminate II, and pick E as the correct answer.

to verify III, suppose a=3, b=10, c=12
c^2= 144
b^2=100
a^2=9
a^2+b^2=109, which is less than c^2. so possible.
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M08-31  [#permalink]

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New post Updated on: 16 Jul 2016, 21:14
I think this is a high-quality question and I agree with explanation.
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Originally posted by rishi02 on 16 Jul 2016, 08:22.
Last edited by rishi02 on 16 Jul 2016, 21:14, edited 1 time in total.
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Re M08-31  [#permalink]

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New post 16 Jul 2016, 11:26
I think this is a high-quality question and I agree with explanation.
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Re: M08-31  [#permalink]

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New post 19 Feb 2017, 14:27
I was reading this post because I went wrong in this question, I am not sure that ration between angles is the reciprocal of the one of the sides since the sin theorem states that: A/sen(opposite)=B/sin(opposite)=C/sin(opposite) and sin is not a linear function
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Re: M08-31  [#permalink]

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New post 22 Jul 2018, 02:26
Easy and very simple solution, thanks team!
Re: M08-31 &nbs [#permalink] 22 Jul 2018, 02:26
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