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Intern  Joined: 21 Nov 2014
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Which side of triangle PQR is the longest?  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 74% (01:29) correct 26% (01:30) wrong based on 775 sessions

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Which side of triangle PQR is the longest?

(1) PQ is longer than QR.
(2) The measure of angle PQR is 70 degrees.
Math Expert V
Joined: 02 Aug 2009
Posts: 7984
Re: Which side of triangle PQR is the longest?  [#permalink]

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livio04 wrote:
Which side of triangle PQR is the longest?

(1) PQ is longer than QR.
(2) The measure of angle PQR is 70 degrees.

Hi,

RULE - the longest side of a triangle has the largest angle opposite to it..

lets see the statements

(1) PQ is longer than QR.
we know QR is not the answer..
But we do not know about the third side PR

Insuff

(2) The measure of angle PQR is 70 degrees
we know from this that the other two angles combined are 180-70=110..
largest side has the largest angle against it..
It is possible that both angles are less than 70 : 60 and 50
Also it is possible that one angle is more than 70 : 75 and 35

Insuff

Combined
we still cannot say if PQ is longest or PR
Insuff

E
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Re: Which side of triangle PQR is the longest?  [#permalink]

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Which side of triangle PQR is the longest?

(1) PQ is longer than QR.
(2) The measure of angle PQR is 70 degrees.

In the original condition, for a triangle, there are 3 variables, which should match with the number of equations. So you need 3 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make E the answer as 1 more equation is needed.
When 1) & 2), you cannot figure out which angle is the biggest and which side is the longest, which is not sufficient.
Thus, the answer is E.

 For cases where we need 3 more equations, such as original conditions with “3 variables”, or“4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.
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Re: Which side of triangle PQR is the longest?  [#permalink]

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chetan2u wrote:
livio04 wrote:
Which side of triangle PQR is the longest?

(1) PQ is longer than QR.
(2) The measure of angle PQR is 70 degrees.

Hi,

RULE - the longest side of a triangle has the largest angle opposite to it..

lets see the statements

(1) PQ is longer than QR.
we know QR is not the answer..
But we do not know about the third side PR

Insuff

(2) The measure of angle PQR is 70 degrees
we know from this that the other two angles combined are 180-70=110..
largest side has the largest angle against it..
It is possible that both angles are less than 70 : 60 and 50
Also it is possible that one angle is more than 70 : 75 and 35

Insuff

Combined
we still cannot say if PQ is longest or PR
Insuff

E

Thanks for the post - what's your rationale for 1)&2)?
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Joined: 06 Dec 2016
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Re: Which side of triangle PQR is the longest?  [#permalink]

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1
Which side of triangle PQR is the longest?

(1) PQ is longer than QR.
(2) The measure of angle PQR is 70 degrees.

(1) clearly insufficient
(2) clearly insufficient
(1)+(2)
Analyze: angle Q <> angle R <> 70 degrees => PQ <>PR
Case 1: angle Q = angle P = 70 degrees => PR = QR
Case 2: angle R < 70 degrees => RQ max
Case 3: angle R > 70 degrees => PR max
=> E
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Re: Which side of triangle PQR is the longest?  [#permalink]

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What about that case where side 1 + side 2 > side 3.
Impossible triangle theorem I think its called.
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Re: Which side of triangle PQR is the longest?  [#permalink]

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akoundourakis wrote:
What about that case where side 1 + side 2 > side 3.
Impossible triangle theorem I think its called.

The rule for triangles : s1-s2<s3<s1+s2.

(1) states that PQ>QR, but we no other information is given- is PR >PQ? or is PR<PQ. If so the longest side changes according to these conditions. Again, no units given, so we can't calculate any thing.
(2) states that angle PQR (basically, the angle opposite side PR) is 70degrees. 180-70= 110, therefore the other 2 angles will have a sum of 110. The problem arises when the angles can be added as 75+35 or 80+30 or 90+20, etc.

Both statements considered together, just lets us know that Angle PQR is 70degrees, which is opposite PR, and we don't know if PR >=<PQ, and PR can be the longest side or medium side.

Hence, E.
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Re: Which side of triangle PQR is the longest?  [#permalink]

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chetan2u Bunuel
If largest side is opposite to smallest angle, in the above question using both statements:
angle PRQ + RPQ=110
Since PQ>QR, angle opposite to PQ is less than angle opp to QR which means PRQ<RPQ which implies PRQ is minimum. This means PQ is the largest side.
Please guide me where am I going wrong?
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Joined: 02 Aug 2009
Posts: 7984
Which side of triangle PQR is the longest?  [#permalink]

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singh8891 wrote:
chetan2u Bunuel
If largest side is opposite to smallest angle, in the above question using both statements:
angle PRQ + RPQ=110
Since PQ>QR, angle opposite to PQ is less than angle opp to QR which means PRQ<RPQ which implies PRQ is minimum. This means PQ is the largest side.
Please guide me where am I going wrong?

Hi..

The longest side has the largest angle opposite to it..
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Re: Which side of triangle PQR is the longest?  [#permalink]

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chetan2u wrote:
singh8891 wrote:
chetan2u Bunuel
If largest side is opposite to smallest angle, in the above question using both statements:
angle PRQ + RPQ=110
Since PQ>QR, angle opposite to PQ is less than angle opp to QR which means PRQ<RPQ which implies PRQ is minimum. This means PQ is the largest side.
Please guide me where am I going wrong?

Hi..

The longest side has the largest angle opposite to it..

Thank you for the reply. Apologies for the insane query Re: Which side of triangle PQR is the longest?   [#permalink] 19 Dec 2018, 08:20
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