Question

If $R_1$ is the circumradius of the pedal triangle of a given triangle $ABC,$ and $R_2$ is the circumradius of the pedal triangle of the pedal triangle formed, and so on $R_3$,$R_4,.....$ then the value of $\sum _{i=1}^{\infty }{R}_i$, where $R$ (circumradius) of $\Delta ABC$ $=5$ is:

• A. $8$
• B. $5$
• C. $12$
• D. $15$

Answer 1

The correct option is B $5$

Given: Circumradius of triangle $ABC,R=5.$
We know, circumradius of pedal triangle, $R_1=\frac{5}{2}$, and so on.
Now $\sum _{i=1}^{\infty }{R}_i=R_1+R_2+R_3+\cdots \infty$
$\Rightarrow \sum _{i=1}^{\infty }{R}_i=\frac{5}{2}+\frac{5}{2^2}+\frac{5}{2^3}+\cdots \infty$
$\Rightarrow \sum _{i=1}^{\infty }{R}_i=\frac{\frac{5}{2}}{1-\frac{1}{2}}=5$