Question

1. in triangle, hij, hi = 48, and ij = 27. what is the range of values for the length third side?

-21 < hj < 75
21 < hj < 48
-21 < hj < 21
21 < hj < 75

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that the length of any side of a triangle must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. For a triangle with sides \(a\), \(b\), and \(c\), \(|a - b| < c < a + b\).

Step2: Identify the given sides

Here, \(a = 48\) (HI) and \(b = 27\) (IJ). Let the third side be \(HJ = c\).

Step3: Calculate the difference and sum

First, find the absolute difference: \(|48 - 27| = 21\). Then, find the sum: \(48 + 27 = 75\).

Step4: Determine the range for the third side

Using the triangle inequality theorem, the third side \(HJ\) must satisfy \(21 < HJ < 75\) (since the length of a side cannot be negative, we ignore the negative part of the absolute difference result as side lengths are positive).

Answer:

21 < HJ < 75 (the option with this inequality, e.g., if it's the last option: "21 < HJ < 75")