The hypotenuse AC of a right-angled ABC is 3\sqrt{5} cm. If AB is doubled and BC is tripled such that ABC remains a right-angled triangle, the hypotenuse becomes 15 cm. What is AB+BC equal to?

Explanation

Let AB = c and BC = a. We have c^2 + a^2 = (3\sqrt{5})^2 = 45. Also, (2c)^2 + (3a)^2 = 15^2 \implies 4c^2 + 9a^2 = 225. Multiplying the first equation by 4 gives 4c^2 + 4a^2 = 180. Subtracting it from the second gives 5a^2 = 45 \implies a^2 = 9 \implies a=3. Then c^2 = 45-9 = 36 \implies c=6. AB+BC = 6+3 = 9 cm.

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