The longest side of a triangle is three times the shortest side. The third side is $2$ cm shorter than the longest side. If the perimeter is at least $61$ cm, what is the **minimum** length of the shortest side?
A$7$ cm
B$9$ cm
C$10$ cm
D$12$ cm
Answer & Solution
Correct answer: B. $9$ cm
Let $x$ be the shortest side. Longest $= 3x$. Third $= 3x - 2$.
Perimeter: $x + 3x + (3x - 2) = 7x - 2 \geq 61 \Rightarrow 7x \geq 63 \Rightarrow x \geq 9$.
Minimum shortest side $= \boxed{9}$ cm.
Check at $x = 9$: sides $9, 27, 25$. Perimeter $= 61$ ✓. Triangle inequality: $9 + 25 = 34 > 27$ ✓.
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