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# In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,

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In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,  [#permalink]

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24 Nov 2018, 10:10
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Question Stats:

74% (03:12) correct 26% (03:38) wrong based on 27 sessions

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GMATbuster's Weekly Quant Quiz#10 Ques #5

In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST, ∠STQ = 60 deg and PT is a prime number greater than 5, then which of the following could be the value of QS?
A) 3
B) 4
C) 14
D) 21
E) 25

Attachment:

Q10.jpg [ 9.3 KiB | Viewed 625 times ]

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Re: In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,  [#permalink]

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24 Nov 2018, 10:33
Since Angle STQ = 60 deg and QS = ST, angle TQS = 60 deg, so the third angle has to be 60 deg, making triangle QTS an equilateral triangle.

In triangle RPT: PR = 3, TR = 5, so third side PT has to satisfy: (5-3) < PT < (5+3) i.e. 2 < PT < 8, but PT is a prime number greater than 5, hence PT has to be 7!

In triangle PTQ: PQ = 15 - 3 = 12, PT = 7, so third side QT has to satisfy: (12-7) < QT < (12+7) i.e. 5 < QT < 19. Since QT = QS, we get 5 < QS < 19.

Only option in this range is 14.

hence Option C
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Re: In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,  [#permalink]

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24 Nov 2018, 10:54
C.
Since ST=QS so angle QTS = angle TQS

And since angle STQ is 60 degree....and other two angle are same so both will be 60 degree and triangle QTS is equilateral triangle.

So QS =QT.

For length of QT
In traingle QTR QR=15 and RT =5.
Since the theorem that sum of two sides is greater than the third side...

Assuming QT is larger side...so QR + RT should be greater than QT

10+5 =15.
So it should be between 10 to 15.
So option C.

In another case if we QR to be longest side in that case sum of QT and RT should be greater than 10.
For that QT should be greater than 5.
Which is not provided in option.

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Re: In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,  [#permalink]

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24 Nov 2018, 11:17
QR = 15 & PR = 3 --> THIS IMPLIES THAT PQ = 12
RT = 5
QS = QT & ANGLE(QTS) = 60 --> THIS IMPLIES THAT TRIANGLE(QTS) IS AN EQUILATERAL TRIANGLE

NOW, SINCE ANY SINGLE SIDE OF A TRIANGLE CANNOT BE GREATER THAN THE SUM OF THE OTHER 2 SIDES --> TP <= 3+5
WE ALSO KNOW THAT TP >5 AND TP IS A PRIME --> TP HAS TO BE 7

AGAIN, USING THE SAME LOGIC, TQ <= 12+7 --> THE ONLY VIABLE OPTION LEFT IS 14
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Re: In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,  [#permalink]

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24 Nov 2018, 11:52
[quote="gmatbusters"]

GMATbuster&#39;s Weekly Quant Quiz#10 Ques #5

In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST, ∠STQ = 60 deg and PT is a prime number greater than 5, then which of the following could be the value of QS?
A) 3
B) 4
C) 14
D) 21
E) 25

Ans is QS =25(E)

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Re: In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,  [#permalink]

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24 Nov 2018, 11:56
[quote="ankitahyd2013"][quote="gmatbusters"]

GMATbuster&#39;s Weekly Quant Quiz#10 Ques #5

In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST, ∠STQ = 60 deg and PT is a prime number greater than 5, then which of the following could be the value of QS?
A) 3
B) 4
C) 14
D) 21
E) 25

Ans is QS =25(E)

Angle STQ=60 deg
And QS=ST
Therefore its an equilateral traiane with all sides equal and all angle equal to 60 degree
Angle TQP = 90 deg
PQ= 12
By pythagorous theorem QT = 25
QT= QS = 25

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Re: In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,  [#permalink]

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24 Nov 2018, 13:59
QR = 15 cm and PR= 3 cm gives QP = 12 ................... (I)

RT = 5 cm and PT is a prime number greater than 5. Using side measure property, measure of any side of triangle lies between positive difference of other 2 sides and sum of other 2 sides, we get RT as 7 because range of RT is between 5-3=2 to 5+3=8 and only number greater than 5 that is prime is 7. So, PT=7 .................. (II)

Now, QS = ST and ∠STQ = 60 deg. Using side-angle property that equal sides in a triangle have equal angles, we get ∠TQS = 60 deg.

We know that sum of angles in a triangle = 180 deg, so third angle ∠TSQ also equals 60 deg.
Hence, this triangle is an equilateral triangle, so all sides are equal and hence QS=ST=QT ............. (III)

Using I and II, QP lies between 12-7=5 and 12+5=17 and using III, QS also lies between 5 an 17.

Of the choices, only 14 falls in this range and hence C is the correct answer.
Re: In the given figure, if QR = 15 cm, PR= 3 cm and RT =5 cm. If QS = ST,   [#permalink] 24 Nov 2018, 13:59
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