Question

1. use patterns and structure what is the minimum side length needed to complete the triangle? explain.

Explanation:

Step1: Recall Triangle Inequality Theorem

For any triangle with side lengths $a$, $b$, $c$, the sum of any two sides must be greater than the third side. Let the unknown side be $x$.

Step2: Apply inequalities to find bounds

First, $5 + 3 > x$ (sum of known sides > unknown), so $x < 8$.
Second, $5 + x > 3$, which simplifies to $x > -2$ (irrelevant, since length is positive).
Third, $3 + x > 5$, which simplifies to $x > 5 - 3$.
<Expression>
$x > 2$
</Expression>

Answer:

The minimum side length needed is just greater than 2 inches. This is because of the Triangle Inequality Theorem, which requires the sum of the two shorter sides of a triangle to be greater than the longest side. For the given sides 5 inches and 3 inches, the unknown side must be longer than $5 - 3 = 2$ inches to satisfy $3 + x > 5$.