Question

in \\( \triangle a b c, a b = 5 \mathrm { ft } \\) and \\( b c = 3 \mathrm { ft } \\). which inequality represents all possible values for the length of \\( a c \\) in feet?
a \\( 2 \leq a c \leq 8 \\)
b \\( 2 < a c < 8 \\)
c \\( 3 \leq a c \leq 7 \\)
d \\( 3 < a c < 7 \\)

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 \(\triangle ABC\), we are given \(AB = 5\) ft and \(BC = 3\) ft. Let \(AC = x\) (the side we want to find the inequality for).

Step3: Apply the triangle inequality

First, find the difference of the two given sides: \(|AB - BC|=|5 - 3| = 2\).
Then, find the sum of the two given sides: \(AB + BC=5 + 3 = 8\).
By the triangle inequality theorem, the length of \(AC\) must satisfy \(|AB - BC| < AC < AB + BC\), which translates to \(2 < AC < 8\).

Answer:

B. \(2 < AC < 8\)