Question

Apply the triangle inequality theorem to determine if size with length of 3‘4 feet and 5 feet form a triangle

Answer 1

The triangle inequality theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must hold:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In this case, you have two sides of lengths \( a = 3.33 \) feet (3 feet 4 inches) and \( b = 5 \) feet. To determine whether these lengths can form a triangle, we need a third side \( c \). However, we can find the range of values for \( c \) that would allow a triangle to be formed with the other two sides.

1. From the first inequality, \( 3.33 + 5 > c \) implies \( c < 8.33 \).

2. From the second inequality, \( 3.33 + c > 5 \) implies \( c > 5 - 3.33 \) or \( c > 1.67 \).

3. From the third inequality, \( 5 + c > 3.33 \) which implies \( c > 3.33 - 5\) or \( c > -1.67\) (this condition is always true since \( c \) must be positive).

Combining these inequalities, the side \( c \) must satisfy:

\[
1.67 < c < 8.33
\]

So, any length \( c \) that falls within the range \( (1.67, 8.33) \) feet will ensure that the lengths of 3 feet 4 inches and 5 feet can indeed form a triangle.