Is the angle [b]a[/b] equal to 30?

Question

GMATBusters’ Quant Quiz Question -5 https://gmatclub.com/forum/download/file.php?id=56246 Is the angle a equal to 30? 1) The length of PQ is twice the length of RQ. 2) angle b = 60 deg WhatsApp Image 2020-07-26 at 7.45.14 AM.jpeg

urshi · Answer

Is the angle a equal to 30?
1) The length of PQ is twice the length of RQ.
NOT SUFFICIENT

Cannot deduce the angle since it is not mentioned if PQ is the diameter or not.

2) angle b = 60 deg

The central angle for the arc ROP will be 120 degree. hence the angle POR will be 30 . This implies the angle a will be greater than 30. Hence SUFFICIENT answer NO to the question.

OPTION B

Varunsawhney8 · Answer

Let Angle PRQ=X

We know,

Angle PRQ= 90 ( Circle Property - Chord Property)

Thus,

90+a+b+c= 180

a+b=90

Statement 1,

The length of PQ is twice the length of RQ.

We know, Cos b= RQ/ PQ = 1/2

We get b= 60 degree

We can easily find value of a from this.

We can calculate a= 30 degree

Sufficient

Statement 2,

angle b = 60 deg

We can calculate a= 30 degree

Sufficient

RajatJ79 · Answer

Is the angle a equal to 30?

Looking at the diagram, line PQ can be diameter or just another chord. If PQ is the diameter, triangle RPQ will be a Right-angled triangle.

Considering  statement 1 alone 
 1) The length of PQ is twice the length of RQ.

If PQ is the diameter, triangle RPQ will be a Right-angled triangle.
In a 30-6-90 Right-angled triangle, the hypotenuse (here PQ) is twice the length of side (here RQ) opposite 30 degree angle (here angle a)

But cannot prove that RPQ is a Right-angled triangle using only statement 1.
Hence,  statement 1 by itself is not sufficient .

Considering  statement 2 alone 
 2) angle b = 60 deg

Statement 2 alone does not prove that RPQ is a 30-6-90 Right-angled triangle
Hence,  statement 1 by itself is not sufficient .

Combining statement 1 and 2 together 
If the hypotenuse (here PQ) is twice the length of side (here RQ) opposite 30 degree angle (here angle a)
and angle b = 60 deg
then, RPQ is a 30-6-90 Right-angled triangle with PQ as the diameter of the circle.

Hence, angle a = 30
 Both statements together are sufficient. 
  Answer is Option C

gkannampGVK · Answer

My Take

- statement A just tells the ratio of the two sides of the triangle. Since we know nothing about the value of the angles we can't do much.

- Statement B says b - 60; Thereby making the sum of the other two angles 120. This alone gives nothing

- Now both A and B together
 PQ/RQ = c/a ( ratio of sides = ratio of sin of opp angles) = 1/2

Archit3110 · Answer

target ; is angle a = 30*
#1
The length of PQ is twice the length of RQ.
using ∆ rule we can say that value of PR ; has to be <3x ; many possible values so PR can be x, 1.5x , √3x or 2x ; insufficient

#2
angle b = 60 deg
since figure is not drawn to scale so its not sufficient to say whether angle PRQ is 90* or not ; hence its insufficient to say
from 1 &2
we know that angle RQP is 60* and and sides ∆ RQ ; x and PQ = 2x and length PR would be x√3 which means PQR is a 30:60:90 ∆ ; hence we can say that angle angle a is 30* hence sufficient
OPTION C

Is the angle a equal to 30?
1) The length of PQ is twice the length of RQ.
2) angle b = 60 deg

njindal · Answer

Answer is "   C   ".

This question can be solved using cosine formula:  c^2 = a^2 + b^2 -2abCosC

Statement 1 : The length of PQ is twice the length of RQ.
This information alone is not sufficient for us to find the value of  a . We need more information.

Statement 2 : angle b = 60 deg
This information alone is not sufficient for us to determine  a . We need more information.

Both Statement Together : The length of PQ is twice the length of RQ and angle b = 60 deg
This information seems enough for us to determine the value of angle P and state if it is  30^{\circ}

Let the sides opposite to angle P, Q, R be p, q and r respectively.
Given: From Statement 1 & 2, r = 2p and b =  60^{\circ}

Step 1: Using cosine formula for  b  (Angle Q),
 q^2 = p^2 + r^2 - 2prCos(b)

Replacing values of r and Q from given, we get (Cos 60 = 1/2)
 q^2 = 3p^2  or,  q = \sqrt{3}p

Step 2: Now using Cosine formula for  a  (Angle P),
 p^2 = q^2 + r^2 - 2qrCos(a)

Replacing values for r and q from given and Step 1, we get
 Cos(a) = \sqrt{3}/2 
Therefore, a =  30^{\circ}

Based on information provided in both statements, we can conclude that the angle  a  is indeed  30^{\circ} .
Therefore, Answer is   C   (Both Statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient)

ashh · Answer

Option C

Both statements together are sufficient but alone are not sufficient.

nothing345 · Answer

Is the angle a equal to 30?
1) The length of PQ is twice the length of RQ.
We don't know whether PQ is the diameter of the circle so the angles can assume any number of values. PQ can equal QR and that changes the measure of the angle.

2) angle b = 60 deg
Again, this doesn't help much when we're not sure about whether PQ is the diameter. 
Could be an equilateral triangle and not a right angle triangle.

So 1 and 2 are insufficient.  combining them doesn't provide any new info.

So the answer is E.

dishra · Answer

Is the angle a equal to 30?
1) The length of PQ is twice the length of RQ.
2) angle b = 60 deg

Statement 1) The length of PQ is twice the length of RQ. 
applying sine rule : ( sin a)/x = (sinc)/2x since a +b+c =180 => c=180-a-b
 
 ( sin a)/x = (sin(180-a-b))/2x => sin a = (sin(a+b))/2 =>  not sufficient

Statement 2) angle b = 60 deg =>  not sufficient

Statement 1+ 2 
sin a = (sin(a+60))/2 => 2sina = sina *cos 60 + cos a* sin60 => 2sina = (sina) *1/2 + (cos a)* √3 /2
3/2 sina = √3 /2 cos a => tana = 1/ √3 => a=30

Answer C

Is the angle [b]a[/b] equal to 30?

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