Question

in triangle qrs, qr = 8 and rs = 5. which expresses all possible lengths of side qs?
○ qs = 13
○ 5 < qs < 8
○ qs > 13
○ 3 < qs < 13

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. Mathematically, for a triangle with sides \(a\), \(b\), and \(c\), \(|a - b| < c < a + b\).

Step2: Identify the given sides

In triangle \(QRS\), we are given \(QR = 8\) and \(RS = 5\). Let \(QS = x\).

Step3: Apply the triangle inequality theorem

First, find the difference of the two given sides: \(|QR - RS| = |8 - 5| = 3\).
Then, find the sum of the two given sides: \(QR + RS = 8 + 5 = 13\).
By the triangle inequality theorem, we have \(3 < x < 13\), which means \(3 < QS < 13\).

Answer:

\(3 < QS < 13\) (the last option: \(3 < QS < 13\))