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Radioactive Triangle

Age 11 to 14 Short Challenge Level:
 
As triangle ABC is equilateral, ∠BAC = $60$°. Since the grey triangles are equilateral, ∠ADE = $60$°, so the triangle ADE is equilateral.

The length of the side of this triangle is equal to the length of $DE = (5 + 2 + 5)\;\mathrm{cm} = 12\;\mathrm{cm}$. So $AF = AD - FD = (12 - 5)\;\mathrm{cm} = 7\;\mathrm{cm}$.

By a similar argument, we deduce that $BD = 7\;\mathrm{cm}$, so the length of the side of the triangle $ABC = (7 + 5 + 7)\;\mathrm{cm} = 19\;\mathrm{cm}$.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.