Question

The radii $r_1,r_2,r_3$ of the escribed circles of the triangle $ABC$ are in H.P. If the area of the triangle is $24{\text{cm}}^2$ and its perimeter is $24\text{cm}$, then the length of its largest side is

• A. $10$
• B. $9$
• C. $8$
• D. $7$

Answer 1

The correct option is A $10$

Given $r_1,r_2,r_3$ are in H.P. Thus,
$\frac{s-a}{△},\frac{s-b}{△},\frac{s-c}{△}$ are in A.P.
Or, $s-a,s-b,s-c$ are in A.P.
Or, $a,b,c$ are in A.P. and
$a+b+c=24=2s$
Or, $2b+b=24,b=8\Rightarrow c=16-a$
Also given $\sqrt{s(s-a)(s-b)(s-c)}=24$
Or $\sqrt{12(12-a)(12-8)(a-4)}=24$
Or $(12-a)(a-4)=12$
OR $a^2-16a+60=0\Rightarrow a=6,10$