Question

what is the value of t? t = submit

Explanation:

Step1: Use the Angle - Bisector and Perpendicular Distance Property

Since \(CA\) is the angle - bisector of \(\angle BCD\) (\(\angle BCA=\angle DCA = 57^{\circ}\)), and \(AB\perp CB\), \(AD\perp CD\). By the Angle - Bisector Theorem (the distances from a point on the angle - bisector to the two sides of the angle are equal), we have \(AB = AD\).

Step2: Set up the equation

Given \(AB = 4t\) and \(AD=t + 69\), we set up the equation \(4t=t + 69\).

Subtract \(t\) from both sides of the equation: \(4t-t=t + 69-t\).

We get \(3t=69\).

Divide both sides by 3: \(t=\frac{69}{3}\).

Answer:

\(23\)