Question

what is the length of \\( \overline{pr} \\) ?

Explanation:

Step1: Identify Similar Triangles

In triangle \(ABC\) and triangle \(PQR\), \(\angle B = \angle Q = 75^\circ\), \(\angle C=\angle R = 55^\circ\), so by AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle PQR\).

Step2: Find the Ratio of Corresponding Sides

In similar triangles, the ratio of corresponding sides is equal. Let's find the ratio of sides \(AB\) and \(PQ\), \(AB = 20\), \(PQ=5\). The ratio \(\frac{AB}{PQ}=\frac{20}{5} = 4\).

Step3: Find the Length of \(PR\)

We know that \(AC = 24\) and \(PR=x\). Since \(\triangle ABC\sim\triangle PQR\), \(\frac{AC}{PR}=\frac{AB}{PQ}\). Substituting the values, \(\frac{24}{x}=4\). Solving for \(x\), we get \(x=\frac{24}{4}=6\).

Answer:

C. 6