Question 25 of 40advanced🔧 ApplyNumerical5 marks

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.

Correct Answer

9 cm

Exercise: EXERCISE 5.1 | Q: 25 | (Page Page 8)
For More Understanding

Explanation

This question from the Linear Inequalities chapter requires setting up an inequality based on the given conditions. The perimeter condition 'at least 61 cm' translates to ≥\geq 61. Students must define variables for all three sides in terms of the shortest side, form the inequality, and solve it. Additionally, for a triangle to exist, the triangle inequality must be satisfied (sum of any two sides greater than the third), which provides a lower bound that must be checked against the solution obtained from the perimeter condition.

Solution Steps

  1. Step 1: Let the shortest side of the triangle be x cm. Then the longest side = 3x cm, and the third side = (3x - 2) cm.

  2. Step 2: Using the perimeter condition (perimeter is at least 61 cm): x + 3x + (3x - 2) ≥\geq 61

  3. Step 3: Simplify: 7x - 2 ≥\geq 61, which gives 7x ≥\geq 63, therefore x ≥\geq 9.

  4. Step 4: Check triangle inequality: For a valid triangle, sum of two shorter sides must exceed the longest side: x + (3x - 2) > 3x, giving x > 2. Since x ≥\geq 9 satisfies x > 2, the triangle is valid.

  5. Step 5: Therefore, the minimum length of the shortest side is 9 cm.

PreviousNext