Question

what is the value of t?
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Explanation:

Step1: Use the Angle - Bisector Theorem (distance from a point to the sides of an angle)

Since \(EG\) is the angle - bisector of \(\angle HEF\), and \(EH\perp HG\), \(EF\perp FG\), then \(EH = EF\).
We know that \(EH = 4t\) and \(EF=t + 66\).

Step2: Set up the equation

Set \(4t=t + 66\).
Subtract \(t\) from both sides: \(4t-t=t + 66-t\).
We get \(3t=66\).

Step3: Solve for \(t\)

Divide both sides of the equation \(3t = 66\) by \(3\): \(t=\frac{66}{3}\).

Answer:

\(22\)