GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 05 Dec 2019, 09:47 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # M08-31

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 59561

### Show Tags

1
9 00:00

Difficulty:   85% (hard)

Question Stats: 46% (01:43) correct 54% (01:35) wrong based on 69 sessions

### HideShow timer Statistics

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
Math Expert V
Joined: 02 Sep 2009
Posts: 59561

### Show Tags

2
2
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 $$c^2&gt;a^2+b^2$$.

Intern  Joined: 22 Aug 2014
Posts: 38

### Show Tags

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.
Board of Directors P
Joined: 17 Jul 2014
Posts: 2492
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
WE: General Management (Transportation)

### Show Tags

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.
Manager  Joined: 21 Sep 2015
Posts: 73
Location: India
GMAT 1: 730 Q48 V42 GMAT 2: 750 Q50 V41 ### Show Tags

I think this is a high-quality question and I agree with explanation.

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.
Manager  B
Joined: 23 Apr 2014
Posts: 57
Location: United States
GMAT 1: 680 Q50 V31 GPA: 2.75

### Show Tags

I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 23 Jan 2017
Posts: 21

### Show Tags

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
Intern  B
Joined: 20 Aug 2016
Posts: 42
GMAT 1: 570 Q46 V23 GMAT 2: 610 Q49 V25 GMAT 3: 620 Q45 V31 WE: Information Technology (Other)

### Show Tags

Easy and very simple solution, thanks team!
Intern  B
Joined: 02 Feb 2018
Posts: 31

### Show Tags

mvictor wrote:
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

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.

I think that's wrong.

Ratio of angles is not always the same as ratio of lengths, for example:
Angles are in the ratio 1:2:3 -> 30-60-90 triangle
lengths of 30-60-90 triangle are in the ratio 1:$$\sqrt{3}$$:2 and not 1:2:3
Director  V
Joined: 12 Feb 2015
Posts: 956

### Show Tags

Given sides a, b, and c to be the lengths of the three sides of triangle, with length c being the longest and a + b > c by the triangle inequality, then if the triangle is obtuse, then $$a^2 + b^2 < c^2$$ and if the triangle is acute, then $$a^2 + b^2 > c^2$$
_________________
________________
Manish "Only I can change my life. No one can do it for me"
Intern  B
Joined: 07 Jan 2018
Posts: 1

### Show Tags

I think this is a high-quality question.
Manager  G
Joined: 22 Jun 2017
Posts: 166
Location: Argentina
Schools: HBS, Stanford, Wharton
GMAT 1: 630 Q43 V34 ### Show Tags

I think this is a high-quality question and I agree with explanation.
_________________
The HARDER you work, the LUCKIER you get.
Intern  B
Joined: 04 Jul 2018
Posts: 1

### Show Tags

I think this is a high-quality question. Re: M08-31   [#permalink] 14 Sep 2019, 06:54
Display posts from previous: Sort by

# M08-31

Moderators: chetan2u, Bunuel  