The longest side of a triangle is three times the shortest side, and the third side is 2 cm shorter than the longest side if the perimeter of the triangle at least 61 cm, Find the minimum length of the shortest-side.

We are given with a triangle,

The longest side of this triangle = 3 × Shortest side …(i)

The third side of this triangle = Longest side – 2 cm …(ii)

The perimeter of the triangle ≥ 61 cm …(iii)

Let

Shortest side of the triangle = a

The longest side of the triangle = b

The third side of the triangle = c

So

From (i),

b = 3 × a

⇒ b = 3a …(iv)

From (ii),

c = b – 2

⇒ c = 3a – 2 (∵ b = 3a) …(v)

Then, perimeter is given by

Perimeter of the triangle = a + b + c

Substituting the values of b and c from equation (iv) and (v) respectively, we get

Perimeter of the triangle = a + (3a) + (3a – 2)

⇒ Perimeter of the triangle = 7a – 2 …(vi)

Putting the value of perimeter of the triangle from (v) in inequality (iii), we get

7a – 2 ≥ 61

⇒ 7a ≥ 61 + 2

⇒ 7a ≥ 63

⇒ a ≥ 9

This means, ‘a’ which is the shortest side of the triangle is 9 or more than 9.

Thus, the minimum length of the shortest side of the triangle is 9 cm.

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