Chapter 1.5, Problem 25PPSE

Solution

To describe the length of the longest side if the perimeter is less than 54 cm.

The longest side is less than 21 cm.

Given:

The lengths of the sides of a triangle are in the ratio $5:6:7$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$, then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$, then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.
• Perimeter of a figure is the sum of all sides.

Calculation:

Let x be the ratio factor of the sides.

Since lengths of the sides of a triangle are in the ratio $5:6:7$, the length of the sides is given by $5x,6x,\text{and }7x$ .

Since the perimeter of the frame is less than 54 cm, it follows

  $\begin{array}{c}P<54\\ 5x+6x+7x<54\\ 18x<54\\ \frac{18x}{18}<\frac{54}{18}\\ x<3\end{array}$

So, the length of the longest side is

  $\begin{array}{c}\text{Longest}\text{side}=7x\\ <7\cdot 3\\ <21\end{array}$

Thus, the length of the longest side of the triangle must be less than 21 cm.