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)
Shortest side of the triangle = a
The longest side of the triangle = b
The third side of the triangle = c
b = 3 × a
⇒ b = 3a …(iv)
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.