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09 May 2019, 06:26
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
31% (03:30) correct
69%
(02:37)
wrong
based on 29
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Consider a rectangle ABCD of area 90 units. The points P and Q trisect AB, and R bisects CD. The diagonal AC intersects the line segments PR and QR at M and N respectively. What is the area of the quadrilateral PQMN ?
A)> 9.5 and ≤ 10
B)> 10 and ≤ 10.5
C)> 10.5 and ≤ 11
D)> 11 and ≤ 11.5
E)> 11.5
A)> 9.5 and ≤ 10
B)> 10 and ≤ 10.5
C)> 10.5 and ≤ 11
D)> 11 and ≤ 11.5
E)> 11.5
Manas1212
In \(\triangle AMP\) and \(\triangle CMR\)
\(\angle MAP\) = \(\angle MCR\)
\(\angle AMP\) = \(\angle CMR\)
Therefore, we can say that \(\triangle AMP\) and \(\triangle CMR\) are similar
Hence, we can say that
\(\frac{AP}{CR}\)=\(\frac{MP}{MR}\)
⇒\(\frac{AB/3}{AB/2}\)=\(\frac{MP}{MR}\)
⇒ MR = \(\frac{3}{2}\) MP
⇒ MR = \(\frac{3}{4}\) RP ... (1)
Similarly, In \(\triangle ANQ\) and \(\triangle CNR\)
\(\angle NAQ\) = \(\angle NCR\)
\(\angle ANQ\) = \(\angle CNR\)
Therefore, we can say that \(\triangle ANQ\) and \(\triangle CNR\) are similar
Hence, we can say that
⇒\(\frac{2AB/3}{AB/2}\)=\(\frac{NQ}{NR}\)
⇒NR = \(\frac{3}{4}\) NQ
⇒NR = \(\frac{3}{7}\) RQ... (2)
In \(\triangle RMN\) and \(\triangle RPQ\)
\(\frac{Area of Triangle RMN}{Area of Triangle RPQ}\)=\(\frac{0.5 RM*RN*Sin MRN}{0.5 RP*RQ*Sin PRQ}\)
So Area of \(\triangle RMN\) = \(\frac{3}{5}\) * \(\frac{3}{7}\) * 15 = \(\frac{27}{7}\)
We know that, Area of triangle RPQ = 1/6*Area of rectangle ABCD = 1/6*90 = 15 sq. units
Hence, the area of the quadrilateral PQNM = 15 - \(\frac{27}{7}\) = \(\frac{78}{7}\) sq. units. Therefore, option D is the correct answer.
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10 Jun 2019, 14:13
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Manas1212
https://drive.google.com/file/d/1xN4GLXSs1HOZHxBhqD_WNGCyFasP2sLq/view?usp=sharing
P & Q are trisecting. So the area of a rectangle with PQ will be 90/3= 30
R is mid of PQ. So outer part of PR and QR will be 30/2= 15
Now we need to reduce area of RMN from triangle RPQ.
MN is diagonal and we can say that those small triangle with M & N are identical. Also area of PQR is equally distributed in 4 parts.
So our final equation will be (15 /4) * 3= 45/4
D is the correct answer.
