Data Sufficiency

[email protected] wrote:If arc PQR above is a semicircle, what is the length of
diameter PR ?
(1) a=4
(2) b=1

We can answer this question without performing any calculations. Instead, we can use some visualization.

Important point: For geometry DS questions, we are typically checking to see whether the statements "lock" a particular angle or length into having just one value. This concept is discussed in much greater detail in our free video: https://www.gmatprepnow.com/module/gmat- ... cy?id=1103

This technique can save a lot of time, however, it is difficult to explain in text only (which is why I encourage you to watch our free video).

Target question: What is the length of diameter PR?
We want to check whether the statements lock this side into having just 1 possible length.

Given: If arc PQR above is a semicircle.
This means that angle PQR is 90 degrees (an important property of circles)

Statement 1: a = 4
If a = 4, then we now have the lengths of 2 sides of a right triangle.
So, we could apply the Pythagorean Theorem to find the length of side PQ.
Since we can find the lengths of all 3 sides of that right triangle, there is only 1 triangle in the universe with those lengths. In other words, statement 1 "locks" the left-hand triangle into exactly 1 shape.
This means that the angle QPR is locked into one angle.
In turn, angle QRP is locked into one angle
So, all three angles of triangle PQR are locked.
Plus we could determine the length of side PQ.
All of this tells us that statement 1 locks triangle PQR into 1 and only 1 triangle, which means there must be only one possible value for the length of side PR.
Since we could (if we chose to perform the necessary calcations) answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: b = 1
If b = 1, then we now have the lengths of 2 sides of a right triangle (the small triangle on the right-hand side).
So, we could apply the Pythagorean Theorem to find the length of side QR.
Since we can find the lengths of all 3 sides of that right triangle, there is only 1 triangle in the universe with those lengths. In other words, statement 2 "locks" the small triangle (on the right side) into exactly 1 shape.
This means that the angle PRQ is locked into one angle.
In turn, angle QPR is locked into one angle
So, all three angles of triangle PQR are locked.
All of this tells us that statement 2 locks triangle PQR into 1 and only 1 triangle, which means there must be only one possible value for the length of side PR.
Since we could (if we chose to perform the necessary calcations) answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent