Question

1. two sides of a triangle have lengths of 13 inches and 18 inches. which of the following could not be the length of its third side? (1) 6 inches (2) 8 inches (3) 13 inches (4) 33 inches 3. two sides of a triangle have lengths of 8.25 meters and 14.75 meters. which of the following could be the length of its third side? (1) 4.5 meters (2) 6.25 meters (3) 18.3 meters (4) 23.5 meters

Explanation:

Step1: Aplicar la desigualdad triangular

La desigualdad triangular establece que para un triángulo con lados \(a\), \(b\) y \(c\), se cumple \(|a - b|\lt c\lt a + b\).

Para el primer problema (\(a = 13\) y \(b=18\)):
\(|13 - 18|=| - 5| = 5\) y \(13 + 18=31\). Entonces \(5\lt c\lt31\).
Para el segundo problema (\(a = 8.25\) y \(b = 14.75\)):
\(|8.25-14.75|=|-6.5| = 6.5\) y \(8.25 + 14.75=23\). Entonces \(6.5\lt c\lt23\).

Step2: Analizar las opciones

• Primer problema:
• Opción (1): \(6\) satisface \(5\lt6\lt31\).
• Opción (2): \(8\) satisface \(5\lt8\lt31\).
• Opción (3): \(13\) satisface \(5\lt13\lt31\).
• Opción (4): \(33\) no satisface \(5\lt33\lt31\).
• Segundo problema:
• Opción (1): \(4.5\) no satisface \(6.5\lt4.5\lt23\).
• Opción (2): \(6.25\) no satisface \(6.5\lt6.25\lt23\).
• Opción (3): \(18.3\) satisface \(6.5\lt18.3\lt23\).
• Opción (4): \(23.5\) no satisface \(6.5\lt23.5\lt23\).

Answer:

1. (4) 33 inches
2. (3) 18.3 meters