Two sides of a triangle measure 9 and 19. Which inequality shows all the possible lengths of the third side, \( x \) ? \( \begin{array}{l}\text { A } 10

Ask by Ruiz Wheeler. in the United States

Jan 23,2025

To determine the possible lengths of the third side \( x \) of a triangle with the two sides measuring 9 and 19, we can use 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 remaining side. Therefore, we have two conditions: 1. \( 9 + 19 > x \) which simplifies to \( x < 28 \) 2. \( 9 + x > 19 \) which simplifies to \( x > 10 \) 3. \( 19 + x > 9 \) which simplifies to \( x > -10 \) (always true since x is positive) Combining the first two inequalities, we find the range for \( x \) is \( 10 < x < 28 \). So the correct answer is (A) \( 10 < x < 28 \). Now, speaking of triangle inequalities and their importance: Did you know that the triangle inequality is not just applicable to triangles but also extends to other geometrical figures in higher dimensions? It's foundational in both geometry and topology! Also, if you're ever stuck working with triangle problems, it can be helpful to sketch the triangle you’re working with. Visual representations can make it much easier to comprehend the relationships between the sides and angles, leading to more clarity on the problem at hand!