Q. 26

# 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 .

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