Question

the figure shows two triangles, where x represents the length of the common side \\( \overline{bd} \\)

(the diagram is not drawn to scale.)

which inequality shows all possible values for x?

a \\( 6 < x < 23 \\)
c \\( 6 < x < 30 \\)
b \\( 3 < x < 30 \\)
d \\( 3 < x < 23 \\)

Explanation:

Step1: Apply triangle inequality to \(\triangle ABD\)

In \(\triangle ABD\), by triangle inequality \(|AB - AD|\lt BD\lt AB + AD\). Given \(AB = 13\) and \(AD=10\), we have \(|13 - 10|\lt x\lt13 + 10\), so \(3\lt x\lt23\).

Step2: Apply triangle inequality to \(\triangle BCD\)

In \(\triangle BCD\), by triangle inequality \(|BC - CD|\lt BD\lt BC + CD\). Given \(BC = 18\) and \(CD = 12\), we have \(|18-12|\lt x\lt18 + 12\), so \(6\lt x\lt30\).

Step3: Find the intersection of the two inequalities

To satisfy both inequalities, we take the intersection of \(3\lt x\lt23\) and \(6\lt x\lt30\). The intersection of these two intervals is \(6\lt x\lt23\).

Answer:

A. \(6\lt x\lt23\)