The lengths of the sides of $\triangle\text{ABC}$ are consecutive integers. It $\triangle\text{ABC}$ has the same perimeter as an equilateral triangle with a side of length $9\ cm$, what is the length of the shortest side of $\triangle\text{ABC}?$
A
$4$
B
$6$
C
$8$
D
10
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C
$8$
Let the sides of $\triangle\text{ABC}$ be $n, n + 1, n + 2$.
$\Rightarrow $ Perimeter $= n + n + 1 + n + 2$
$\Rightarrow (9 + 9 + 9) = 3n + 3$
$\Rightarrow 3n = 24$
$\Rightarrow n = 8cm$
Thus, the shortest side is $8\ cm$.
Hence, correct option is $(c)$.
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Each side of a triangle is multiplied with the sum of the squares of the other two sides. If the sum of all such possible results is 6 times the product of the sides, then the triangle must be