Question

In triangle, $ABC$, if $r_1=3,r_2=10,r_3=15,$ then $c=$

• A. $5$
• B. $12$
• C. $13$
• D. $13/2$

Answer 1

Answer: (C)

$13$

Given: $r_1=3,r_2=10,r_3=15$

Now we have given the values of $r_1,r_2$and $r_3$, so in solution of triangles,

we know that:

$\frac{1}{r}=\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}$

$⟹\frac{1}{r}=\frac{1}{3}+\frac{1}{10}+\frac{1}{15}$

$⟹\frac{1}{r}=\frac{10+3+2}{3}$

$⟹r=2$

Also, we know that:

${\Delta }^2=r\times r_1\times r_2\times r_3$

${\Delta }^2=2\times 3\times 10\times 15$

${\Delta }^2=900$

$\Delta =\sqrt{900}$

$\Delta =30$

Now,$r=\frac{\Delta }{s}$

$⟹s=\frac{\Delta }{r}$

$⟹s=\frac{30}{2}=15$

$r_3=\frac{\Delta }{(15-c)}$

$⟹15=\frac{30}{(15-c)}$

$⟹15-c=2$

$⟹c=13$

Hence the value of c is $13$.