Question

The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.

Answer 1

Given:

Let the shortest side of the triangle be x cm.

Then,
Longest side = 3x cm
Third side = (3x − 2) cm

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Step 1: Form the perimeter inequality

Perimeter ≥ 61 cm

x + 3x + (3x − 2) ≥ 61

7x − 2 ≥ 61

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Step 2: Solve the inequality

7x ≥ 63

x ≥ 9

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Step 3: Check triangle inequality

Sum of two smaller sides > longest side

x + (3x − 2) > 3x

4x − 2 > 3x

x > 2

This condition is satisfied for x ≥ 9.

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Final Answer:

The minimum length of the shortest side of the triangle is
9 cm.