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23 Jan 2020, 01:59
Kudos
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A
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Difficulty:
25%
(medium)
Question Stats:
76% (01:59) correct
24%
(02:22)
wrong
based on 82
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Points A, B, and, C have xy-coordinates (2,0), (8,12), and (14,0), respectively. Points X, Y, and Z have xy-coordinates (6,0), (8,4), and (10,0), respectively. What fraction of the area of triangle ABC is the area of triangle XYZ?
(A) 1/9
(B) 1/8
(C) 1/6
(D) 1/5
(E) 1/3
(A) 1/9
(B) 1/8
(C) 1/6
(D) 1/5
(E) 1/3
Archit3110
Major Poster
23 Jan 2020, 08:03
Kudos
Bookmarks
Bunuel
for ∆ ABC ; the base = 12 and height ; 12 ; area ; 1/2 * 12* 12 ; 72
for ∆ XYX ; base ; 4 and height ; 4 ; area ; 1/2 * 4*4 ; 8
ratio ; 8/72 ; 1/9
IMO A
29 Jan 2020, 05:53
Kudos
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Bunuel
For triangle ABC, we see that two of the vertices, (2, 0) and (14, 0), are actually on the x-axis, so 14 - 2 = 12 can be the base of the triangle. Then we can use the (absolute value of the) y-coordinate of the remaining vertex as the height. In this case, it is 12 since the remaining vertex is (8, 12). Therefore, the area of triangle ABC is ½ x 12 x 12 = 72.
Similarly, for triangle XYZ, we see that the base is 10 - 6 = 4 and the height is 4. Therefore, the area of triangle XYZ is ½ x 4 x 4 = 8.
So the area of triangle XYZ is 8/72 = 1/9 of the area of triangle ABC.
Answer: A
