Question

In $\Delta ABC$, $r = 1$, $R = 4$, $\Delta = 8$ then the value of $ab + bc + ac = $
A) 18
B) 81
C) 72
D) 27

Answer 1

Hint: To solve this question we should know about some formulas which are ${r_1} + {r_2} + {r_3} = 4R + r$ and $r\left( {{r_1} + {r_2} + {r_3}} \right) = ab + bc + ac - {s^2}$, where ${r_1},{r_2}$ and ${r_3}$ are ex-radii of the circumcircle and $a,b,c$ are sides of the triangle.

Complete step-by-step answer:
We have given that the circumradius of a circle is $R = 4$, in-radius of the same circle is $r = 1$ and the area of that same triangle is $\Delta = 8$ and we have to find the value of $ab + bc + ac$.
First, we calculate the semi perimeter of the given triangle because it is required in the solution.
We have a formula of in-radius $r = \dfrac{\Delta }{s}$ by which we will calculate the semi perimeter. Where, $r$ is in-radius, $\Delta $ is the area of the triangle and $s$ is the semi perimeter of the triangle.
Now, we will substitute the values of inradius and area in this formula to find the semi perimeter.
$\begin{gathered}
  1 = \dfrac{8}{s} \\
  s = 8 \\
\end{gathered} $
So, the semi perimeter of the given triangle is 8
Now, we will solve for the sum of ex-radii which are ${r_1} + {r_2} + {r_3} = 4R + r$ .
Now substitute the value of circumradius and inradius in this formula to find the value of the sum of ex-radii.
$\begin{gathered}
  {r_1} + {r_2} + {r_3} = 4\left( 4 \right) + \left( 1 \right) \\
   = 17 \\
\end{gathered} $
Now, we have got the sum of ex-radii.
Use the relation between the sides of the triangle, in-circle radius, and ex-radii of the triangle, which is given as:
$r\left( {{r_1} + {r_2} + {r_3}} \right) = ab + bc + ac - {s^2}$ to find $ab + bc + ac$
$r\left( {17} \right) = ab + bc + ac - {s^2}$
Now, substitute the given value of inradius and obtained value of semi perimeter in the above equation find the sum of $ab + bc + ac$
$\begin{gathered}
  \left( 1 \right)\left( {17} \right) = ab + bc + ac - {\left( 8 \right)^2} \\
  17 = ab + bc + ac - 64 \\
  17 + 64 = ab + bc + ac \\
  81 = ab + bc + ac \\
\end{gathered} $
So, we got the required value of $ab + bc + ac$ which is 81.

Therefore, option B is the correct answer.

Note: Semi perimeter of the triangle is half of the perimeter of the triangle. If the sides of the triangle are $a,b$ and $c$, then the semi-perimeter of the triangle is given as:
$s = \dfrac{{a + b + c}}{2}$