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The hypotenuse of a right-triangle is $3 \sqrt{10} cm$ If the smaller side is tripled and thelonger side is doubled, the new hypotenuse will be $9 \sqrt{5} cm .$ Find the length of each

side.

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3,9

4,7

8,9

0,6

answer is A.

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Detailed Solution

Suppose the smaller leg = 'a'
longer leg ='b'.
Hypotenuse

^{2} = {length}^{2} + {breadth}^{2}

Given, the length of hypotenuse of right triangle =

3 \sqrt{10} cm

So, by the Pythagoras theorem,

{(3 \sqrt{10})}^{2} = a^{2} + b^{2}

9 \times 10 = a^{2} + b^{2}

\Rightarrow a^{2} + b^{2} = 90 \dots \dots . . (1)

Now, the smaller leg is tripled while the longer leg is doubled.
Thus, the new hypotenuse =

9 \sqrt{5} cm

\begin{matrix} \Rightarrow (3 a)^{2} + (2 b)^{2} = 81 \times 5 \\ \Rightarrow 9 a^{2} + 4 b^{2} = 405 \dots \dots \dots (2) \end{matrix}

Multiple equation (1) by 4 and subtract from eq 2 to get,

\Rightarrow 5 a^{2} = 45

\Rightarrow a^{2} = 9

\Rightarrow a = 3

Thus,

9 + b^{2} = 90

\Rightarrow b^{2} = 81

\Rightarrow b = 9

Correct option is 1.

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