Question

if ( mangle pqs = mangle rqs = 43^{circ} ) and ( ps = 24 ), what is ( rs )?

Explanation:

Step1: Apply Angle - Bisector Theorem for Right - Angles

The Angle - Bisector Theorem for angles that are part of right - angled triangles (in this case, \(\angle SPQ = 90^{\circ}\) and \(\angle SRQ=90^{\circ}\)) states that if a ray bisects an angle of a triangle and is perpendicular to the opposite side, then the two segments formed by the bisector on the opposite side are equal. Here, \(QS\) is the angle - bisector (\(m\angle PQS=m\angle RQS = 43^{\circ}\)), \(PS\perp PQ\), and \(RS\perp RQ\).

Step2: Use the property of angle - bisector in right - angled triangles

Since \(QS\) is the angle - bisector of \(\angle PQR\), \(PS\perp PQ\), and \(RS\perp RQ\), by the property of the angle - bisector in the case of right - angled triangles (a special case of the Angle - Bisector Theorem for two right - angled triangles \(\triangle PQS\) and \(\triangle RQS\) sharing the common side \(QS\)), we have \(PS = RS\).

Answer:

\(24\)