Question

If a=13 , b=15 , c=15 , show tgat R=65/8 , r=4 , r1=21/2 , r2=12 and r3=14.​

Answer 1

Given:

Given values are

$a=13\\b=14\\c=15$

To find:

We should find the values of $R,r,r_1,r_2$ and $r_3$.

Solution:

We know that,

$s=\frac{a+b+c}{2} \\\triangle =\sqrt{s(s-a)(s-b)(s-c)} \\ R=\frac{abc}{4\triangle} \\r=\frac{\triangle}{s}\\r_1=\frac{\triangle}{s-a}\\r_2=\frac{\triangle}{s-b}\\r_3=\frac{\triangle}{s-c}$

First, find the value of $s$.

Substitute the required values in the formula.

$\Rightarrow s=\frac{a+b+c}{2}$

$\\\Rightarrow s=\frac{13+14+15}{2}$

$\\\Rightarrow s=\frac{42}{2}$

$\\\Rightarrow s=21$

Now, find the value of $\triangle$.

Substitute the required values in the formula.

$\Rightarrow \triangle =\sqrt{s(s-a)(s-b)(s-c)}$

$\Rightarrow \triangle =\sqrt{21(21-13)(21-14)(21-15)}$

$\Rightarrow \triangle =\sqrt{21(8)(7)(6)}$

$\Rightarrow \triangle =\sqrt{7056}$

$\Rightarrow \triangle =84$

Now, we will find the values of $R,r,r_1,r_2$ and $r_3$.

Finding the value of $R$,

Substitute the required values in the formula.

$\Rightarrow R=\frac{abc}{4\triangle}$

$\Rightarrow R=\frac{13\times14\times15}{4\times84}$

$\Rightarrow R=\frac{2730}{336}$

$\Rightarrow R=\frac{65}{8}$

Finding the value of $r$,

Substitute the required values in the formula.

$\Rightarrow r=\frac{\triangle}{s}$

$\Rightarrow r=\frac{84}{21}$

$\Rightarrow r=4$

Finding the value of $r_1$,

Substitute the required values in the formula.

$\Rightarrow r_1=\frac{\triangle}{s-a}$

$\Rightarrow r_1=\frac{84}{21-13}$

$\Rightarrow r_1=\frac{84}{8}$

$\Rightarrow r_1=\frac{21}{2}$

Finding the value of $r_2$,

Substitute the required values in the formula.

$\Rightarrow r_2=\frac{\triangle}{s-b}$

$\Rightarrow r_2=\frac{84}{21-14}$

$\Rightarrow r_2=\frac{84}{7}$

$\Rightarrow r_2=12$

Finding the value of $r_3$,

Substitute the required values in the formula.

$\Rightarrow r_3=\frac{\triangle}{s-c}$

$\Rightarrow r_3=\frac{84}{21-15}$

$\Rightarrow r_3=\frac{84}{6}$

$\Rightarrow r_3=14$

Therefore, the values are $R=\frac{65}{8} , r=4 , r_1=\frac{21}{2} , r_2=12$ and $r_3=14$.