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22 Nov 2010, 06:27
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A
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Difficulty:
5%
(low)
Question Stats:
86% (01:17) correct
14%
(01:30)
wrong
based on 1332
sessions
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In the rectangular coordinate system above, the area of triangle PQR is what fraction of the area of triangle LMN ?
A. 1/9
B. 1/8
C. 1/6
D. 1/5
E. 1/3
Attachment:
Coordinate.jpg [ 25.98 KiB | Viewed 58192 times ]
22 Nov 2010, 07:43
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In the rectangular coordinate system above the area of triangle PQR is what fraction of the area triangle LMN?
A. 1/9
B. 1/8
C. 1/6
D. 1/5
E. 1/3
The area of a triangle equals to \(area=\frac{1}{2}*base*height\).
Now the base of triangle PQR (or simply the length of line segment PR) is 4 units (the difference between x-coordinates of the points P and R), and the height is also 4 units (the altitude from the point Q to the x-axis equals to 4 units). So \(area_{PQR}=\frac{1}{2}*4*4=8\);
Similarly the base of triangle LMN is 12 units (the difference between x-coordinates of the points L and N), and the height is also 12 units (the altitude from the point M to the x-axis equals to 12 units). So \(area_{LMN}=\frac{1}{2}*12*12=72\);
Thus \(\frac{area_{PQR}}{area_{LMN}}=\frac{8}{72}=\frac{1}{9}\).
Answer: A.
As for the difficulty level, I'd say around 600.
Hope it's clear.
29 Jul 2013, 03:32
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monirjewel
Understanding the question:
Two triangles are given. They look proportional/similar but this should be confirmed. Also, means are provided to calculate length of base and height.
Facts to refer:
If 2 triangles are similar, then the ratio of their area= (ratio of the sides)^2
What's given in the question and what it implies (noted as =>):
Coordinates for L, P, R, N are given => Base of smaller triangle = 4 and that of larger triangle =12
Coordinate for M and Q are given => Height of smaller triangle = 4 and that of larger triangle =12
What is asked for:
Area of PQR/Area of LMN => Ratio of the areas
Solution:
Since the base and height of the triangles are of the same ratio, the 2 triangles are similar. (Since the base and height are of equal ratio, the other 2 sides will also be of the same ratio.) Hence the fact given in "Facts to refer" can be used.
Ratio of sides = 4/12 =1/3
(Ratio of areas) = (1/3)^2 = 1/9
