Question

A teacher asked Rohan to draw a triangle with the following condition: The longest side of the triangle is 7 cm less than twice the shortest side and the third side is 7 cm shorter than the longest side. The perimeter of the triangle is at least 84 cm.

Based on the above information, form a linear inequation and answer the following questions:

1. What is the minimum length of the shortest side? 
2. What is the minimum length of the longest side? 
3. Identify the type of triangle that Rohan has drawn along with the length of the possible sides he got. 
4. What is the least area of the triangle drawn?

Answer 1

Let the shortest side of triangle be x cm.

Given,

Longest side of triangle = 2x − 7

Third side of triangle = (2x − 7) − 7 = 2x − 14

(i) Given,

Perimeter of triangle is atleast 84 cm.

⇒ x + (2x − 7) + (2x − 14) ≥ 84

⇒ 5x − 21 ≥ 84

⇒ 5x ≥ 84 + 21

⇒ 5x ≥ 105

⇒ x ≥ `105/5`

⇒ x ≥ 21.

∴ Minimum length of the shortest side of triangle = 21 cm.

(ii) Length of the longest side of triangle = 2x − 7

Minimum length of longest side will be when x = 21, substituting value we get :

⇒ 2(21) − 7

⇒ 42 − 7

⇒ 35 cm.

∴ Minimum length of the longest side of triangle = 35 cm.

(iii) Minimum length of third side:

Third side of triangle = 2x − 14

= 2(21) − 14

= 42 − 14

= 28 cm.

Three sides of triangle = 21 cm, 28 cm and 35 cm.

⇒ 21² + 28²

= 441 + 784

= 1225

⇒ 35² = 1225.

Thus, we can say that 21² + 28² = 35².

Thus, it is a right angled triangle with hypotenuse 35 cm and other two sides are 21 cm and 28 cm.

∴ The triangle is right-angled triangle.

(iv) By formula,

Area of right angled triangle = `1/2` × product of sides containing right angle

Least area of the triangle = `1/2`​ × 21 × 28

= 294 cm².

∴ The least area of the triangle = 294 cm².