Answer:
28 cm.
Explanation:
To find the length of the longest side of the triangle, we need to determine the values of the side lengths.
The ratio of the side lengths is given as 5:6:7. Let's assume the common ratio is x.
1. Length of the first side:
The length of the first side can be expressed as 5x.
2. Length of the second side:
The length of the second side can be expressed as 6x.
3. Length of the third side:
The length of the third side can be expressed as 7x.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
To find the longest side, we need to consider the maximum value of the sum of the lengths of the two smaller sides.
Let's calculate the maximum perimeter possible when the sum of the lengths of the two smaller sides is less than 54 cm.
(5x + 6x) < 54
11x < 54
x < 4.91
Since x represents the common ratio, it must be a positive value.
Let's consider the nearest whole number less than 4.91, which is 4.
Substituting x = 4 into the expressions for the side lengths, we get:
First side: 5x = 5(4) = 20 cm
Second side: 6x = 6(4) = 24 cm
Third side: 7x = 7(4) = 28 cm
Among these side lengths, the longest side is 28 cm.