The hypotenuse of a right triangle is 3√5cm. If the smaller side is tripled and the longer side doubled, new hypotenuse will be 9√5 cm. How long are the sides of the triangle?

Let the shortest side(AC) be ‘(X)’cms, and the longer side (AB) be ‘(Y)’ cms, as demonstrated in the figure drawn below:

∴ BC = 3√10 cms

∵ (BC)² = (AC)² + (AB)²

∴ (3√10)² = X² + (Y)²

On simplifying further,

(Y)² = 90 – X²

X² + Y² – 90 = 0 –––––– (i)

As per the question,

New smaller side = ‘(3X)’ cms

New longer side = ‘(2Y)’ cms

∴ BC = 9√5 cms

∵ (BC)² = (AC)² + (AB)²

∴(9√5)² = (3X)² + (2Y)²

On simplifying further,

4Y² + 9X² = 405

4X² + 9Y² – 405 = 0 –––––– (ii)

Putting the value of X², from equation (i) in equation (ii)

X² = 90 – Y²

On simplifying further,

(360–4Y)² + 9Y² – 405 = 0

5Y² = 45

Y = ± 3 cms(Only positive values),

∴ X = √(90–9).

∴ X = 9 cms

the shortest side (AC) is X = 9 cms hypotenuse i.e., (BC) is cms and the other side (AB) is 3 cms .