PRMO 2014 question 10

10. In a triangle ABC, X and Y are points on the segments AB and AC, respectively, such that AX : XB = 1 : 2 and AY : YC = 2 : 1. If the area of triangle AXY is 10 then what is the area of triangle ABC? [PRE-RMO–2014]

Solution (using a construction trick):

Let's try to construct a simple triangle satisfying these conditions.
Let's put right angle at A.
Now area of AXY = zk = 10
Area of ABC = 9zk/2 = 45

Solution(using area formula):
One of the area formula of triangle is Area = 1/2(side1 length)(side 2 length)(sin(included angle))
Here the included angle is A which is common in both. So Ratio of area = ratio of sides multiplied.
So area ABC = 3/2 * 3/1 * 10 = 45

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