Sarah is creating a triangular birthday card for her teacher. In order to glue lace around the outside of the card, Sarah needs to know the length measure 4 inches and 9 inches. If the length of the third side is \( x \) inches, which inequality is true? \( 5 \leq x \leq 13 \) \( 5

Ask by Mckenzie Medina. in the United States

Jan 23,2025

To determine the length of the third side of a triangle given two sides, we can apply the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. In this case, if we denote the two sides as 4 inches and 9 inches, we find that the third side \( x \) must satisfy \( 4 + 9 > x \) (which simplifies to \( x < 13 \)), and also \( 4 + x > 9 \) (which simplifies to \( x > 5 \)). Therefore, the correct inequality is \( 5 < x < 13 \). To visualize this, think about the sides of a triangle unlike anything else! If one side were too long or too short, you wouldn't be able to connect all three points to create that lovely triangular shape. Designing your card is much like balancing a triangle—you want to keep all the sides in harmony!