Question

The area of a right triangle is $6864$ sq. units. If the ratio of its legs is $143:24$,

then the value of $\left[\frac{r}{4}\right]$, where $[\cdot ]$ represents the greatest integer function, is?

Answer 1

Let the height be $143k$
Hence breath will be $24k$.
Now area
$=\frac{1}{2}.143\times 24.k^2$
$=6864$
Or
$\frac{24k^2}{2}=48$
Or
$k=2$ ...(since sides have to be positive).
Now
$h=286$ and $b=48$
Hence hypotenuse will be
$=290$.
Thus
$r=\frac{△}{s}$
$=\frac{2△}{a+b+c}$
$=\frac{2\left(6864\right)}{290+286+48}$
$=\frac{1141}{52}$.
Hence
$\frac{r}{4}=\frac{1141}{208}$
$=5.48$ ...(approx).
Hence
$\left[r/4\right]=5$.