In a right angled triangle, the ratio of the shorter sides is 2:3. Find the ratio of the hypotenuse of the triangle to its perimeter. $(A s s u m e \sqrt{13} = 3.5)$

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Solution

Consider right triangle ABC.

Since AB : BC = 2:3, let AB = 2x and BC = 3x.

Using Pythagoras theorem,

$A C^{2} = A B^{2} + B C^{2}$

$\Rightarrow A C^{2} = (2 x)^{2} + (3 x)^{2} = 13 x^{2}$

$\Rightarrow A C = \sqrt{13} x = 3.5 x = \frac{7}{2} x$

Now, perimeter of the triangle

= AC + AB + BC

$= 2 x + 3 x + \frac{7}{2} x = \frac{17}{2} x$

$R e q u i r e d r a t i o = \frac{h y p o t e n u s e}{p e r i m e t e r}$

$= \frac{\frac{7}{2} x}{\frac{17}{2} x} = \frac{7}{17}$

Therefore, ratio of hypotenuse to perimeter is $7 : 17.$

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