To find the length of the longest side of the triangle given the ratio of the sides and the condition on the perimeter, we can start by denoting the sides of the triangle as follows:
- Let the sides be represented as 5x, 6x, and 7x, where x is a common multiplier.
The perimeter of the triangle can be expressed as the sum of the lengths of its sides:
- Perimeter = 5x + 6x + 7x = 18x
The problem states that the perimeter is less than 54 cm:
- 18x < 54
Now, we can solve for x:
- x < 54 / 18
- x < 3
Since x must be less than 3, we need to find the length of the longest side, which is 7x:
- Length of the longest side = 7x
To determine the value of the longest side at its maximum, we can use the highest value of x that meets the condition:
- If x = 3 (the highest possible for calculation purposes), the length of the longest side would be:
- 7 * 3 = 21 cm
However, since x must be less than 3, the actual longest side is slightly under 21 cm. Thus, we can conclude:
- Therefore, the length of the longest side must be less than 21 cm.