Question

if $\angle qrt \cong \angle srt$ and $st = 4$, what is $qt$?
$qt = \square$

Explanation:

Step1: Apply the Angle - Bisector Theorem for Right - Angled Triangles

Since \( \angle QRT=\angle SRT\), \(RT\) is the angle bisector of \( \angle QRS\). Also, \( \angle S = \angle Q=90^{\circ}\).
By the Angle - Bisector Theorem (in the form for right - angled triangles where if a ray bisects an angle of a triangle and is perpendicular to the opposite side, the two segments formed are equal), we know that if a point is on the bisector of an angle and is equidistant from the sides of the angle.
In this case, \(T\) is on the bisector of \( \angle QRS\), and \(TS\perp SR\), \(TQ\perp QR\).

Step2: Conclude the length of \(QT\)

We have the property that \(QT = ST\) (because a point on the angle bisector of an angle is equidistant from the sides of the angle). Given \(ST = 4\).

Answer:

\(4\)