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8. Sequences and Series
medium
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 ?
A
$4$
B
$6$
C
$7$
D
$8$
(IIT-2017)
Solution
Let sides be $\mathrm{a}-\mathrm{d}, \mathrm{a}, \mathrm{a}+\mathrm{d},(\mathrm{d}>0)$
$\Rightarrow \quad a^2+(a-d)^2=(a+d)^2$
$\Rightarrow \quad a=4 d$
$\Rightarrow \quad \text { sides are } 3 d, 4 d, 5 d$
$\text { As area is } 24$
$\Rightarrow \quad \frac{1}{2} \times 3 d \times 4 d=24$
$\Rightarrow \quad d=2$
$\Rightarrow \quad \text { sides are } 6,8,10$
$\Rightarrow \quad \text { smallest side is } 6.$
Standard 11
Mathematics
