Walkabout
If arc PQR above is a semicircle, what is the length of diameter PR ?
(1) a = 4
(2) b = 1
Attachment:
Semicircle.png
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 the video below.
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: 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 = 1If 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
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