If arc PQR above is a semicircle, what is the length of diameter PR ?
(1) a = 4
(2) b = 1
OA D
Source: Official Guide
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If arc PQR above is a semicircle, what is the length of diam
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BTGmoderatorDC
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If arc PQR above is a semicircle, what is the length of diameter PR ?
(1) a = 4
(2) b = 1
OA D
Source: Official Guide
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GMATGuruNY
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An INSCRIBED ANGLE is formed by two chords.BTGmoderatorDC wrote:
If arc PQR above is a semicircle, what is the length of diameter PR ?
(1) a = 4
(2) b = 1
Thus, angle PQR is an inscribed angle.
An inscribed angle that intercepts the diameter is a RIGHT ANGLE.
Thus, angle PQR is a right angle, implying that triangle PQR is a RIGHT TRIANGLE.
In the figure above, PS is a height drawn through right angle PQR.
A height drawn through the right angle of a triangle forms SIMILAR TRIANGLES.
Proof:
If angle QPR = x and angle PQS = y, then x+y = 90.
Since angle PQR = 90, angle SQR = 90-y = x.
Since angle QSR = 90, angle SQR = x, and x+y=90, angle QRP = y.
Thus, all three triangles -- PQS, QRS and PQR -- have the SAME COMBINATION OF ANGLES, as shown in the figure above:
x - y - 90.
Triangles that have the same combination of angles are SIMILAR.
The legs of similar triangles are in the SAME RATIO.
Thus, in all 3 triangles:
(leg opposite x) : (leg opposite y) = (leg opposite x) : (leg opposite y).
In triangle PQS, (leg opposite x) : (leg opposite y) = 2/a.
In triangle QRS, (leg opposite x) : (leg opposite y) = b/2.
Since the two ratios are equal, we get:
2/a = b/2
ab = 4.
Statement 1: a=4
Since ab=4, b=1, implying that PR = 4+1 = 5.
SUFFICIENT.
Statement 2: b=1
Since ab=4, a=4, implying that PR = 4+1 = 5.
SUFFICIENT.
The correct answer is D.
Problems that test the same concept:
https://www.beatthegmat.com/inscribed-tr ... 74152.html
https://www.beatthegmat.com/length-of-th ... 71979.html
https://www.beatthegmat.com/geo-question ... nta-14-649
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BTGmoderatorDC wrote:
If arc PQR above is a semicircle, what is the length of diameter PR ?
(1) a = 4
(2) b = 1
OA D
Source: Official Guide
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 = 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
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