The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24 , th

Question

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24 , then what is the length of its smallest side?

Professor PC Thomas Classes · Answer

(6) : Let the sides be given by $a-d$, $a, a+d$, where $(a, d>0)$. Also, $d<a$.\ By the condition, $$ \begin{aligned} & a^{2}+(a-d)^{2}=(a+d)^{2} \ \Rightarrow \quad & a^{2}=(a+d)^{2}-(a-d)^{2} \Rightarrow a^{2}=4 a d \ 	herefore \quad & a=4 d \end{aligned} $$ Thus the sides are $3 d, 4 d, 5 d$.\ As area $=24 \Rightarrow \frac{1}{2} \cdot 3 d \cdot 4 d=24 	herefore d=2$\ The sides are $6,8,10$.\