Question

if ( ur = 9 ), find ( qu ).

Explanation:

Step1: Use the property of congruent triangles

Since \(PR = 16\), \(PS=13\), \(OQ = 19\) and \(OR = 16\), \(SQ = 13\), \(OT=19\), we can observe that the lines from the vertices \(O\), \(P\), \(Q\) intersect at \(U\). By the property of congruent segments (if two sides of two triangles are equal and the included angles are equal, then the triangles are congruent). Here, we can consider the triangles formed by the segments.

Step2: Apply the congruent - segment property for \(UR\) and \(QU\)

We know that in such a geometric configuration (where the segments from the vertices satisfy the congruence conditions as shown by the equal side - lengths \(PR = OR = 16\), \(PS=SQ = 13\), \(OT=TQ = 19\)), the segments \(UR\) and \(QU\) are equal. Given \(UR = 9\), so \(QU=UR\).

So, \(QU = 9\).

Answer:

$9$