The lengths that make the sense for the value of b are 0.5 inches and 2 inches.
<h2>Given </h2>
An isosceles triangle has two sides of equal length, a, and a base, b.
The perimeter of the triangle is 15.7 inches, so the equation to solve is 2a + b = 15.7.
<h3>What is the triangle inequality theorem?</h3>
If two sides of a triangle are unequal, the longer side has a greater angle opposite to it.
![\rm a+a > b\\\\2a > b](https://tex.z-dn.net/?f=%5Crm%20a%2Ba%20%3E%20b%5C%5C%5C%5C2a%20%3E%20b)
The equation of the perimeter of the triangle;
![\rm 2a+b = 15.7\\\\2a = 15.7-b\\\\a = \dfrac{15.7-b}{2}](https://tex.z-dn.net/?f=%5Crm%202a%2Bb%20%3D%2015.7%5C%5C%5C%5C2a%20%3D%2015.7-b%5C%5C%5C%5Ca%20%3D%20%5Cdfrac%7B15.7-b%7D%7B2%7D)
1. Substitute b = 0.5 in the equation;
![\rm a = \dfrac{15.7-b}{2}\\\\ a = \dfrac{15.7-0.5}{2} \\\\ a = \dfrac{15.2}{2}\\\\a = 7.6](https://tex.z-dn.net/?f=%5Crm%20a%20%3D%20%5Cdfrac%7B15.7-b%7D%7B2%7D%5C%5C%5C%5C%20%20a%20%3D%20%5Cdfrac%7B15.7-0.5%7D%7B2%7D%20%5C%5C%5C%5C%20a%20%3D%20%5Cdfrac%7B15.2%7D%7B2%7D%5C%5C%5C%5Ca%20%3D%207.6)
Then,
Verify the triangle inequality theorem;
0.5 + 7.6 > 7.6 is true
7.6 + 7.6 > 0.5 is true
2. Substitute b = 2 in the equation;
![\rm a = \dfrac{15.7-b}{2}\\\\ a = \dfrac{15.7-02}{2} \\\\ a = \dfrac{13.7}{2}\\\\a = 6.85](https://tex.z-dn.net/?f=%5Crm%20a%20%3D%20%5Cdfrac%7B15.7-b%7D%7B2%7D%5C%5C%5C%5C%20%20a%20%3D%20%5Cdfrac%7B15.7-02%7D%7B2%7D%20%5C%5C%5C%5C%20a%20%3D%20%5Cdfrac%7B13.7%7D%7B2%7D%5C%5C%5C%5Ca%20%3D%206.85)
Then,
Verify the triangle inequality theorem;
0.5 + 76.85 > 6.85 is true
6.85 + 6.85 > 2 is true
Hence, the lengths that make the sense for the value of b are 0.5 inches and 2 inches.
To know more about the Triangle inequality theorem click the link given below.
brainly.com/question/1026055