A triangle is divided into four regions by lines parallel to . The lines divide into equal segments.
If the second largest region has area , find the area of .
A diagram if you need it:
Source: AIMO 2017
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By the AAA (angle-angle-angle) similarity test, we can say that triangles A B 1 C 1 , A B 2 C 2 , A B 3 C 3 , A B C are similar.
Since the lines divide A B into 4 equal segments, the sides of the triangles are in the ratio 1 : 2 : 3 : 4 (i.e. Side Ratio). Therefore their areas are in the ratio 1 : 4 : 9 : 1 6 .
We know that region B 3 C 3 C 2 B 2 has area 2 2 5 . Thus if we let the area of triangle A B 1 C 1 = x , then
2 2 5 = A B 3 C 3 − A B 2 B 2 = 9 x − 4 x = 5 x
x = 5 2 2 5 = 4 5
Therefore the area of triangle A B C is 1 6 x = 1 6 × 4 5 = 7 2 0
The answer is 720 .