Q. 274.7( 3 Votes )

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.

Answer :

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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