2017 AMC 12B Problems/Problem 15
Let be an equilateral triangle. Extend side beyond to a point so that . Similarly, extend side beyond to a point so that , and extend side beyond to a point so that . What is the ratio of the area of to the area of ?
Solution 1: Law of Cosines
Solution by HydroQuantum
Recall The Law of Cosines. Letting , Since both and are both equilateral triangles, they must be similar due to similarity. This means that .
Therefore, our answer is .
Solution 2 (inspection)
Note that the height and base of are respectively 4 times and 3 times that of . Therefore the area of is 12 times that of .
By symmetry, . Adding the areas of these three triangles and for the total area of gives a ratio of , or .