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

RELATED VIDEO

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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

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

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!

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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