Question

In $△ABC$, the incircle touches the sides $BC,CA$ and $AB$ at $D,E$ and $F$ respectively and its radius is $4$ units. If the lengths $BD,CE$ and $AF$ are consecutive integers, then-

• A. Sides are also consecutive integers
• B. Perimeter of the triangle is $42$ units
• C. Diameter of circumcircle os $65$ units
• D. Area of triangle is $84$ sq. units

Answer 1

Answer: (A), (B), (D)

The correct options are
A Sides are also consecutive integers
B Perimeter of the triangle is $42$ units
D Area of triangle is $84$ sq. units

Let $BD=n-1,CE=n$ and $AF=n+1$
Then $BC=2n-1,AC=2n+1$ and $AB=2n$
$\Rightarrow$ $s=\frac{AB+BC+AC}{2}=3n$
$\Delta =\sqrt{3n(n+1)(n-1)n}$
$\therefore r=\frac{\Delta }{s}=\frac{\sqrt{3n^2(n^2-1)}}{3n}=\sqrt{\frac{n^2-1}{3}}=4\Rightarrow n=7$
So, the sides of the triangle are $13,14,15$ and perimeter $=13+14+15=42$
$\Delta =rs=4.21=84$ sq.units
$R=\frac{abc}{4\Delta }=\frac{\mathrm{13.14.15}}{4.84}=\frac{65}{8}$