Points M and P lie on square LNQR, and LM = PQ. What is the length of the line segment PQ?
(1) PR = (4/3)√10
(2) The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1
Statement 1: PR = (4/3)√10
Since QR=4, PR = (4/3)√10, and PQ² + QR² = PR², we can solve for PQ.
SUFFICIENT.
Statement 2: The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1.
Since the two statements cannot contradict each other, the case given in statement 1 -- QR=4 and PR = (4/3)√10 -- must also satisfy statement 2.
Check whether this is the ONLY case that will satisfy statement 2.
Plugging QR=4 and PR = (4/3)√10 into PQ² + QR² = PR², we get:
PQ² + 4² = [(4/3)√10]²
PQ² + 16 = 160/9
PQ² = 160/9 - 144/9
PQ = 4/3.
In this case:
Area of ∆PQR = (1/2)(PQ)(QR) = (1/2)(4/3)(4) = 8/3.
Since LM = PQ, ∆LMN = ∆PQR = 8/3.
Thus:
The total area of the shaded region = ∆LMN + ∆PQR = 8/3 + 8/3 = 16/3.
The total area of the unshaded region = LNQR - shaded region = 16 - 16/3 = 32/3.
Resulting ratio:
unshaded : shaded = (32/3) : (16/3) = 2:1.
The value given in statement 1 -- PR = (4/3)√10 -- yields the 2:1 ratio required by statement 2.
If the length of PR increases or decreases, the ratio of the two areas will NOT be 2:1.
Thus, to satisfy statement 2, it must be true that PR = (4/3)√10, implying that PQ = 4/3.
SUFFICIENT.
The correct answer is
D.
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