Question

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. $(Assume\sqrt{13}=3.5)$

Answer 1

Consider right triangle ABC.

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

Using Pythagoras theorem,

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

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

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

Now, perimeter of the triangle

= AC + AB + BC

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

$Requiredratio=\frac{hypotenuse}{perimeter}$

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

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