In the figure above, what is the perimeter of ΔPQR ?

Question

https://gmatclub.com/forum/download/file.php?id=50479 In the figure above, what is the perimeter of ΔPQR ? (1) The length of segment PT is 2. (2) The length of segment RS is \sqrt{3} . DS34402.01 2019-09-22_0537.png

BrentGMATPrepNow · Accepted Answer

Target question :    What is the perimeter of ΔPQR ?

Statement 1 : The length of segment PT is 2  
Since angles in a triangle add to 180°, we know that ∠PQT = 45°, which means ΔPQT is a  right isosceles triangle . 
This means QT = 2 and from there we can use the Pythagorean theorem to show that PQ = 2√2 
 https://i.imgur.com/jQ1vTJ7.png

At this point, if we focus our attention on the blue right triangle below, we see that we have a  30-60-90 special right triangle 
So when we compare the blue right triangle with the purple base 30-60-90 right triangle, we can determine the lengths of the remaining sides to get: 
  https://i.imgur.com/f3gTOnB.png

We now get the following measurements:
 https://i.imgur.com/15fjuZI.png

So, the answer to the target question is  the perimeter of ΔPQR = 2√2 + 4 + 2√3 + 2 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : The length of segment RS is  \sqrt{3}  
 IMPORTANT : For geometry Data Sufficiency questions, we're typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the video below

Notice that, when we add RS = √3 the diagram, we can still mentally move points P and Q (in the directions specified in the diagram below) so that the 30° angle at point R is preserved and the 45° angle at point P is preserved. 
 https://i.imgur.com/HoQYB7r.png 
Since we are able to move points P and Q, we can conclude that  the perimeter of ΔPQR can have many possible values.  
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

RELATED VIDEO
 https://www.youtube.com/watch?v=wetqSGMdY7Y

shridhar786 · Answer

In the figure above, what is the perimeter of ΔPQR?

STATEMENT (1) The length of segment PT is 2.
--PQT is an isosceles triangle so PQ = 2 \sqrt{2} 
in QTR (30-60-90 triangle) --QT = 2 then QR = 4 RT = 2 \sqrt{3} 
---perimeter of triangle PQR = 2+4+2+2 \sqrt{3}  = 8+2 \sqrt{3} 
SUFFICIENT

STATEMENT (2) The length of segment RS is √3.
from this statement, we can't find the value of PQ, QR, and PR
so, we can't find the perimeter
INSUFFICIENT

A is the correct answer

coreyander · Answer

The highlighted needs correction. 
Perimeter =  2+4+2\sqrt{2}+2\sqrt{3} = 6+ 2\sqrt{2}+2\sqrt{3}

Jsound996 · Answer

If you know Angle/Side/Angle or ASA ( is when we know two angles and a side between the angles), we can find the other sides of the triangle and hence the perimeter of the triangle.
Information that we know:
Angle P = 45 degrees 
Angle R = 30 degrees
Triangle PTQ is a right isosceles triangle. (ratio of sides is x: x: x√2)

We can infer that Angle Q = 105 degrees.
So if we know the length of any side of the triangle PQR, we can figure out the perimeter.

(1) The length of segment PT is 2.
We know that one side (x) = 2, so side PQ = 2√2. That is enough information to find the perimeter. (No need to do the math)
 Statement 1 is Sufficient!

(2) The length of segment RS is  \sqrt{3} .
This gives no information of the sides.
 Statement 2 is insufficient!

Answer is A

CEdward · Answer

I don't understand the answers.

St2 -...

QTR is a right triangle so if we know RS = √3 then we know that's half of x√3... (b/c TQR is 60 = 180 - 90 - 30)
x√3 / 2 = √3 ---> x = 2

So now we know TR = 2√3 and from that we can figure out QT. QTP is clearly a 45-45-90 triangle so if we know one side, then we can figure out the others.

Hence, sufficient.

What am I doing wrong here???

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In the figure above, what is the perimeter of ΔPQR ?

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GMAT Data Sufficiency (DS) Questions

In the figure above, what is the perimeter of ΔPQR ?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards