Question

if $\angle qrt \cong \angle srt$, $qt = a + 50$, and $st = 3a$, what is the value of $a$?
$a = \square$

Explanation:

Step1: Use the Angle - Bisector Theorem for Right - Angles

Since \( \angle QRT\cong\angle SRT\), \(RT\) is the angle - bisector. Also, \( \angle S=\angle Q = 90^{\circ}\). By the Angle - Bisector Theorem (in the case of right - angled triangles where the bisector of an angle and the perpendiculars from a point on the bisector to the sides of the angle are considered), \(QT = ST\).

Step2: Set up the equation

We know that \(QT=a + 50\) and \(ST = 3a\). Since \(QT=ST\), we set up the equation \(a + 50=3a\).

Step3: Solve the equation

Subtract \(a\) from both sides of the equation:
\(a+50 - a=3a - a\)
\(50 = 2a\)
Divide both sides by 2: \(a=\frac{50}{2}\)

Answer:

\(25\)