Question

1. the lengths of the sides of a triangle are in the ratio 5:6:7. describe the length of the longest side if the perimeter is less than 54 cm.

Explanation:

Step1: Let the side - lengths

Let the side - lengths of the triangle be $5x$, $6x$, and $7x$ (where $x$ is a positive real number).

Step2: Calculate the perimeter

The perimeter $P$ of the triangle is $P=5x + 6x+7x=18x$.

Step3: Set up the inequality

Since the perimeter is less than $54$ cm, we have the inequality $18x\lt54$.

Step4: Solve the inequality for $x$

Dividing both sides of the inequality $18x\lt54$ by $18$, we get $x\lt3$.

Step5: Find the length of the longest side

The longest side of the triangle is $7x$. Since $x\lt3$, then $7x\lt7\times3 = 21$.

Answer:

The length of the longest side is less than 21 cm.