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

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



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:


We now get the following measurements:


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.

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

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