Question

In a triangle $ABC$ with usual notation, if $r=1,R=3$ and $s=5,$ then which of the following is (are) CORRECT?

• A. Area of triangle $ABC$ is $5$ units
• B. Product of the sides of triangle$ABC$ is $60$
• C. $a^2+b^2+c^2=24$
• D. Sum of the ex-radii of triangle $ABC$ is $13$

Answer 1

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

The correct options are
A Area of triangle $ABC$ is $5$ units
B Product of the sides of triangle$ABC$ is $60$
C $a^2+b^2+c^2=24$
D Sum of the ex-radii of triangle $ABC$ is $13$

$r=\frac{\Delta }{s}$
$\Rightarrow \Delta =rs=5$

Since $s=5\Rightarrow a+b+c=10$
$\Delta =\frac{abc}{4R}$
$\Rightarrow abc=60$

${\Delta }^2=s(s-a)(s-b)(s-c)\Rightarrow 5=(5-a)(5-b)(5-c)\Rightarrow 5=125-25(a+b+c)+5(ab+bc+ca)-abc$
$\therefore ab+bc+ca=38$
Now, $a^2+b^2+c^2=(a+b+c{)}^2-2(38)=24$

Also, $r_1+r_2+r_3-r=4R$
$\therefore r_1+r_2+r_3=r+4R=13$