Question

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.

Explanation:

Step1: Find the range of the third side

According to the triangle - inequality theorem, for a triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Let \(a = 20\) and \(b=14\). Then \(|20 - 14|\lt c\lt20 + 14\), which simplifies to \(6\lt c\lt34\).

Step2: Find the range of the perimeter

The perimeter \(P=a + b + c\). Substitute \(a = 20\) and \(b = 14\) into the perimeter formula. So \(P=34 + c\). Since \(6\lt c\lt34\), then \(34+6\lt P\lt34 + 34\), which means \(40\lt P\lt68\).

Answer:

No. Because the perimeter \(P\) of the triangle satisfies \(40\lt P\lt68\). There is a possibility that the perimeter \(P\) is greater than \(60\) (for example, if \(c = 30\), then \(P=20 + 14+30=64\gt60\)). So she is not guaranteed to have enough fencing.