Question

applications

1. tess is designing a garden in the shape of a triangle. she knows that she wants one side to have a length of 20 feet and another side to have a length of 14 feet. she would like place fencing along each side of the garden. if she has 60 feet of fencing, is she guaranteed to have enough to surround the garden? justify.

image of triangle abc with ab=14 ft, ac=20 ft
third side of the garden to have an integer length,

Explanation:

Step1: Use triangle inequality theorem

Let the sides of the triangle be \(a = 14\), \(b=20\), and \(c\). The triangle - inequality theorem states that \(|a - b|\lt c\lt a + b\). So, \(|14 - 20|\lt c\lt14 + 20\), which simplifies to \(6\lt c\lt34\).

Step2: Find the range of the perimeter

The perimeter \(P=a + b + c=14 + 20 + c=34 + c\). Since \(6\lt c\lt34\), then \(34+6\lt P\lt34 + 34\), or \(40\lt P\lt68\).

Answer:

No. The perimeter \(P\) of the triangle satisfies \(40\lt P\lt68\). While it is possible that the perimeter is less than or equal to \(60\) (for example, if \(c = 26\), \(P=14 + 20+26 = 60\)), it is not guaranteed. Because if \(c\) is close to \(34\) (say \(c = 33\)), then \(P=14 + 20+33=67>60\).