Question

4.) determine a range for the missing side of the triangle below.
triangle with sides 3 (ab), 5 (bc), and ac as the missing side

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for any triangle, the length of any side must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. Let the missing side be \( x \) (side \( AC \)). The two known sides are \( 3 \) and \( 5 \).

Step2: Calculate the lower bound

The lower bound is the absolute difference of the two known sides. So, \( |5 - 3| = 2 \). Thus, \( x > 2 \).

Step3: Calculate the upper bound

The upper bound is the sum of the two known sides. So, \( 5 + 3 = 8 \). Thus, \( x < 8 \).

Answer:

The range for the missing side is \( 2 < x < 8 \)