Question

1. two sides of an isosceles triangle measures 10 and 10. what is true about the third side?

○ < 10
○ > -10
○ < 11
○ < 20

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For a triangle with sides \(a\), \(b\), and \(c\), we have \(a + b>c\), \(a + c>b\), and \(b + c>a\).

Step2: Apply the Theorem to the Given Triangle

In this isosceles triangle, two sides are \(a = 10\) and \(b = 10\). Let the third side be \(c\). Using the triangle inequality theorem, we consider the sum of the two equal sides: \(a + b>c\). Substituting the values of \(a\) and \(b\), we get \(10+10 > c\), which simplifies to \(20>c\) or \(c < 20\). We also need to consider the other inequalities, but since the side length must be positive, and the other inequalities (\(10 + c>10\) simplifies to \(c>0\) and \(10 + c>10\) also simplifies to \(c>0\)) are already satisfied by the positive length and the main inequality from the sum of the two equal sides gives the upper bound.

Answer:

\(\boldsymbol{<20}\) (The option corresponding to \(<20\))